The Annals of Probability 2016, Vol. 44, No. 3, 2349–2425 DOI: 10.1214/15-AOP1023 © Institute of Mathematical Statistics, 2016

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES
BY JI OON LEE∗,1, KEVIN SCHNELLI†,2,5, BEN STETLER‡,3 AND HORNG-TZER YAU‡,4,5
KAIST∗, IST Austria† and Harvard University‡
We consider N ×N random matrices of the form H = W +V where W is a real symmetric or complex Hermitian Wigner matrix and V is a random or deterministic, real, diagonal matrix whose entries are independent of W . We assume subexponential decay for the matrix entries of W , and we choose V so that the eigenvalues of W and V are typically of the same order. For a large class of diagonal matrices V , we show that the local statistics in the bulk of the spectrum are universal in the limit of large N.

1. Introduction. A prominent class of random matrix models is the Wigner ensemble, consisting of N × N real symmetric or complex Hermitian matrices, W = (wij ), whose matrix entries are random variables that are independent up to the symmetry constraint W = W ∗. The first rigorous result about the spectrum of random matrices of this type is Wigner’s global semicircle law [60], which
states that the empirical distribution of the rescaled eigenvalues, (λi), of a Wigner matrix W is given by

(1.1)

1N

1

N

δλi (E)
i=1

−→

ρsc(E)

:=

2π

4 − E2 +

(E ∈ R),

as N → ∞, in the weak sense. The distribution ρsc is called the semicircle law. Let pWN (λ1, . . . , λN ) denote the joint probability density of the (unordered)
eigenvalues of W . If the entries of the Wigner matrix W are i.i.d. (independent
and identically distributed) real or complex Gaussian random variables, the joint density of the eigenvalues, pWN ≡ pGN , is given by

(1.2)

pGN (λ1, . . . , λN ) =

1 ZGN

|λi
i<j

− λj |β e−βN

N i=1

λ2i /4

,

Received June 2014; revised February 2015. 1Supported in part by National Research Foundation of Korea Grant 2011-0013474 and TJ Park
Junior Faculty Fellowship. 2Supported by ERC Advanced Grant RANMAT, No. 338804, and the “Fund for Math.” 3Supported by NSF GRFP Fellowship DGE-1144152. 4Supported in part by NSF Grant DMS-13-07444 and Simons investigator fellowship. 5This work was completed while the authors were visiting the Institute for Advanced Study. MSC2010 subject classifications. 15B52, 60B20, 82B44.
Key words and phrases. Random matrix, local semicircle law, universality.
2349

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LEE, SCHNELLI, STETLER AND YAU

with β = 1, 2, for the real, complex case, respectively. The normalization ZGN ≡ ZGN (β) in (1.2) can be computed explicitly. The real and complex Gaussian matrix ensembles so defined are known as the Gaussian orthogonal ensemble (GOE,
β = 1) and Gaussian unitary ensemble (GUE, β = 2), respectively, and as noted above we denote the corresponding joint densities as pGN instead of pWN .
The n-point correlation functions are defined by

WN ,n(λ1, . . . , λn) := RN−n pWN (λ1, λ2, . . . , λN ) dλn+1 dλn+2 · · · dλN ,
1 ≤ n ≤ N . Using orthogonal polynomials the correlation functions of the GUE and GOE have been explicitly computed by Dyson, Gaudin and Mehta; see, for example, [44]. For the Gaussian unitary ensemble, their results assert that the limiting behavior on small scales at a fixed energy E in the bulk of the spectrum, that is, for |E| < 2, satisfies

(1.3)

1 [ρsc(E)]n

N G,n

E + α1 , E + α2 , . . . , E + αn

ρsc(E)N

ρsc(E)N

ρsc(E)N

−→ det

K (αi

− αj )

n i,j

=1,

as N → ∞, where K is the sine-kernel

K(x, y)

:=

sin π(x − y) π(x − y) .

Note that the limit in (1.3) is independent of the energy E as long as E is in the bulk of the spectrum. The rescaling by a factor 1/N of the correlation functions in (1.3) corresponds to the typical separation of consecutive eigenvalues, and we refer to the law under such a scaling as local statistics. Similar but more complicated formulas were also obtained for the Gaussian orthogonal ensemble; see, for example, [1, 44] for reviews. Note that the limiting correlation functions do not factorize, reflecting the fact that the eigenvalues remain strongly correlated in the limit of large N .
The Wigner–Dyson–Gaudin–Mehta conjecture, or bulk universality conjecture, states that the local eigenvalue statistics of Wigner matrices are universal in the sense that they depend only on the symmetry class of the matrix, but are otherwise independent of the details of the distribution of the matrix entries. The bulk universality can be formulated in terms of weak convergence of correlation functions or in terms of eigenvalue gap statistics. This conjecture for all symmetry classes has been established in a series of papers [22–24, 28, 31, 33]. After this work began, parallel results were obtained for complex Hermitian matrices and certain symmetric matrices in [55, 56]; see [30] for a more detailed review.
In the present paper, we consider deformed Wigner matrices. A deformed Wigner matrix, H , is an N × N random matrix of the form

(1.4) H = V + W,

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2351

where V is a real, diagonal, random or deterministic matrix and W is a real symmetric or complex Hermitian Wigner matrix independent of V . The matrices are normalized so that the eigenvalues of V and W are order one. If the entries, (vi), of V are random we may think of V as a “random potential”; if the entries of V are deterministic, matrices in the form of (1.4) are sometimes referred to as Wigner matrices with external source.
Assuming that the empirical eigenvalue distribution of V ,

ν

:=

1 N

δvi ,
i=1

converges weakly, respectively, weakly in probability, to a nonrandom measure, ν, it was shown in [46] that the empirical distribution of the eigenvalues of H converges weakly in probability to a deterministic measure. This measure depends on ν and is thus in general distinct from ρsc. We refer to it as the deformed semicircle law, henceforth denoted by ρfc. There is no explicit formula for ρfc in terms of ν. Instead, ρfc is obtained as the solution of a functional equation for its Stieltjes transform; see (2.9) below. It is known that ρfc admits a density [6]. Depending on ν, ρfc may be supported on several disjoint intervals. For simplicity, we assume below that ν is such that ρfc is supported on a single bounded interval. Further, we choose ν such that all eigenvalues of H remain close to the support of ρfc; that is, there are no “outliers” for N sufficiently large.
If W belongs to the GUE, H is said to belong to the deformed GUE. The deformed GUE for the special case when V has two eigenvalues ±a, each with equal multiplicity, has been treated in a series of papers [2, 8, 9]. In this setting the local eigenvalue statistics of H can be obtained via the solution to a Riemann– Hilbert problem; see also [17] for the case when V has equispaced eigenvalues. Bulk universality for correlation functions of the deformed GUE with rather general deterministic or random V has been proved in [51] by means of the Brezin– Hikami/Johansson integration formula.
In the present paper, we establish bulk universality of local averages of correlation functions for deformed Wigner matrices of the form H = V + W , where W is a real symmetric or complex Hermitian Wigner matrix and V is a deterministic or random real diagonal matrix. We assume that the entries of W are centered independent random variables with variance 1/N whose distributions decay subexponentially; see Definition 2.1. If V is random, we assume for simplicity that its entries (vi) are i.i.d. random variables. We assume that ν converges weakly, respectively, weakly in probability, to a nonrandom measure ν; see Assumption 2.2. We further assume that the corresponding deformed semicircle law ρfc is supported on a single compact interval and has square root decay at both endpoints. Sufficient conditions for these assumptions to hold have appeared in [52] and are rephrased in Assumption 2.3. Under these assumptions, our main results in Theorem 2.5 and in Theorem 2.6 assert that the limiting correlation functions of the deformed Wigner

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LEE, SCHNELLI, STETLER AND YAU

ensemble are universal when averaged over a small energy window. Note that our
results hold for complex Hermitian and real symmetric deformed Wigner matrices.
Before we outline our proofs, we recall the notion of β-ensemble or log-gas
which generalizes the measures in (1.2). Let U be a real-valued potential, and consider the measure on RN defined by the density

(1.5)

μNU (λ1, . . . , λN )

:=

1 ZUN

i<j

|λi

−

λj |β e−βN

N i=1

(λ2i /2+U

(λi

))/2

,

where β > 0 and ZUN ≡ ZUN (β) is a normalization. Bulk universality for β-ensembles asserts that the local correlation functions for measures in the form of (1.5) are
universal (for sufficiently regular potentials U ) in the sense that for each value of
β > 0 they agree with the local correlation functions of the Gaussian ensemble with U ≡ 0.
For the classical values β ∈ {1, 2, 4}, the eigenvalue correlation functions of μNU can be explicitly expressed in terms of polynomials orthogonal to the exponential
weight in (1.5). Thus the analysis of the correlation functions relies on the asymp-
totic properties of the corresponding orthogonal polynomials. This approach, ini-
tiated by Dyson, Gaudin and Mehta (see [44] for a review), was the starting point for many results on the universality for β-ensemble with β ∈ {1, 2, 4} [7, 18–20, 37, 42, 43, 49].
For general β > 0, bulk universality of β-ensembles has been established in [10–12] for potentials U ∈ C4. Recently, alternative approaches to bulk universality for β-ensembles with general β have been presented in [50] and [4] under
different conditions on U .
We emphasize at this point that the eigenvalue distributions of the deformed
ensemble in (1.4) are in general not of the form (1.5), even when W belongs to the
GUE or the GOE.
Returning to the random matrix setting, we recall that the general approach to
bulk universality for (generalized) Wigner matrices in [24, 28, 33] consists of three
steps:

(1) establish a local semicircle law for the density of eigenvalues; (2) prove universality of Wigner matrices with a small Gaussian component by
analyzing the convergence of Dyson Brownian motion to local equilibrium; (3) compare the local statistics of Wigner ensembles with Gaussian divisible en-
sembles to remove the small Gaussian component of step (2).

For an overview of recent results and this three-step strategy, see [30]. Note that the “local equilibrium” in step (2) refers to measure (1.2), with β = 1, 2, respectively, in the real symmetric, complex Hermitian case.
For deformed Wigner matrices, the local deformed semicircle law, the analogue
of step (1), was established in [39] for random V . However, when V is random, the eigenvalues of V + W fluctuate on scale N −1/2 in the bulk (see [39]), but their

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2353

gaps remain rigid on scale N −1. To circumvent the mesoscopic fluctuations of the eigenvalue positions, we condition on V , considering its entries to be fixed. The methods of [39] can be extended, as outlined in Section 3, to prove a local law on the optimal scale for “typical” realizations of random as well as deterministic potentials V .
Our corresponding version of step (2), a proof of bulk universality for deformed Wigner ensembles with small Gaussian component, is the main novelty of this paper. The local equilibrium of Dyson Brownian motion in the deformed case is unknown but may effectively be approximated by a “reference” β-ensemble that we explicitly construct in Section 4. In Section 5, we analyze the convergence of the local distribution of the deformed Wigner ensemble under Dyson Brownian motion to the “reference” β-ensemble. However, since the “reference” β-ensemble is not given by the invariant GUE/GOE, it also evolves in time. Using the rigidity estimates for the deformed ensemble established in step (1) and the rigidity estimates for general β-ensembles established in [12], we obtain, in Section 5, bounds on the time evolution of the relative entropy between the two measures being compared. The idea to estimate the entropy flow of the Dyson Brownian motion with respect to the “global equilibrium state” given by the GUE/GOE was initiated in [28] and [29]. On the other hand, the idea to use “time dependent local equilibrium states” to control the entropy flow of hydrodynamical equations was introduced in [61]. There it is observed that the change of relative entropy is negligible provided that the time dependent local equilibrium is chosen in agreement with the density predicted by the hydrodynamical equations. In this paper, we combine both methods to yield an effective estimate on the entropy flow of the Dyson Brownian motion in the deformed case. This global entropy estimate is then used in Section 6 to conclude that the local statistics of the locally-constrained deformed ensemble with small Gaussian component agree with those of the locallyconstrained reference β-ensemble. Relying on the main technical result of [31], we further conclude that the local statistics of the locally-constrained reference β-ensemble agrees with the local statistics of the GUE/GOE. Once this conclusion is obtained for the locally-constrained ensembles, it can be extended to the nonconstrained ensembles. This completes step (2) in the deformed case.
In Sections 7 and 8, we outline step (3) for deformed Wigner matrices; the proof is similar to the argument for Wigner matrices in [32]. The main technical input is a bound on the resolvent entries of H on scales N −1−ε that can be obtained from the local law in step (1). In Section 8, we then combine steps (1)–(3) to conclude the proof of our main results, Theorems 2.5 and 2.6.
We remark that our arguments in step (2) do not rely on V being diagonal. Step (3) depends only on the deformed local semicircle law of step (1); in principle, step (3) is independent of whether or not V is diagonal, as long as a deformed local semicircle law is given. Currently, our proof for the deformed local semicircle law uses that V is diagonal.

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LEE, SCHNELLI, STETLER AND YAU

In Section 9, we prove that, in addition to bulk universality, the edge universality also holds for our model, that is, that the local statistics at the spectral edges are given by the Tracy–Widom–Airy statistics. From the main technical result of [12], the proof of the edge universality follows the same three-step program as the proof of bulk universality. A detailed discussion of our edge universality result, Theorem 2.10, and related results can be found in Section 2.4.
In the Appendix, we collect several technical results on the deformed semicircle law and its Stieltjes transform. Some of these results have previously appeared in [52] and [39, 40].

2. Assumptions and main results. In this section, we list our assumptions and our main results.

2.1. Definition of the model. We first introduce real symmetric and complex Hermitian Wigner matrices.

DEFINITION 2.1. A real symmetric Wigner matrix is an N × N random matrix, W , whose entries, (wij ) (1 ≤ i, j ≤ N ), are independent (up to the symmetry constraint wij = wji ) real centered random variables satisfying

(2.1)

Ewi2i

=

2 N

,

Ewi2j

=

1 N

(i = j ).

In case (wij ) are Gaussian random variables, W belongs to the Gaussian orthogonal ensemble (GOE).
A complex Hermitian Wigner matrix is an N × N random matrix, W , whose entries, (wij ) (1 ≤ i, j ≤ N ), are independent (up to the symmetry constraint wij = w¯ ji ) complex centered random variables satisfying

(2.2)

Ewi2i

=

1 N

,

E|wij |2

=

1 N

,

Ewi2j = 0 (i = j ).

For simplicity, we assume that the real and imaginary parts of (wij ) are independent for all i, j . This ensures that Ewi2j = 0 (i = j ). In case (Re wij ) and
(Im wij ) are Gaussian random variables, W belongs to the Gaussian unitary en-

semble (GUE).

Irrespective of the symmetry class of W , we assume that the entries (wij ) have

a subexponential decay, that is,

(2.3)

√ P N |wij | > x

≤ C0e−x1/θ ,

for some positive constants C0 and θ > 1. In particular,

(2.4)

E|wij

|p

≤

C

(θp)θp N p/2

(p ≥ 3).

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2355

Let V = diag(vi) be an N × N diagonal, random or deterministic matrix, whose entries (vi) are real-valued. We denote by ν the empirical eigenvalue distribution of the diagonal matrix V = diag(vi),

(2.5)

ν

:=

1 N

N
δvi .
i=1

ASSUMPTION 2.2. There is a (nonrandom) centered, compactly supported probability measure ν such that the following holds:

(1) If V is a random matrix, we assume that (vi) are independent and identically distributed real random variables with law ν. Further, we assume that (vi) are independent of (wij ).
(2) If V is a deterministic matrix, we assume that there is α0 > 0, such that for any fixed compact set D ⊂ C+ (independent of N ) with dist(D, supp ν) > 0, there is C such that

(2.6)

max dν(v) − dν(v) ≤ CN −α0,

z∈D v − z

v−z

for N sufficiently large.

Note that (2.6) implies that ν converges to ν in the weak sense as N → ∞. Also note that condition (2.6) holds for large N with high probability for 0 < α0 < 1/2 if (vi) are i.i.d. random variables.

2.2. Deformed semicircle law. The deformed semicircle can be described in terms of the Stieltjes transform: for a (probability) measure ω on the real line we define its Stieltjes transform, mω, by

mω(z) :=

dω (v ) v−z

z ∈ C+ .

Note that mω is an analytic function in the upper half plane and that Im mω(z) ≥ 0, Im z > 0. Assuming that ω is absolutely continuous with respect to Lebesgue measure, we can recover the density of ω from mω by the inversion formula

(2.7)

ω (E )

=

lim
η0

1 π

Im

mω(E

+

iη)

(E ∈ R).

We use the same symbols to denote measures and their densities. Moreover, we have

lim
η0

Re

mω(E

+

iη)

=

−

ω(v) dv v−E

(E ∈ R),

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LEE, SCHNELLI, STETLER AND YAU

whenever the left-hand side exists. Here the integral on the right is understood as
principal value integral. We denote in the following by Re mω(E) and Im mω(E) the limiting quantities

(2.8)

Re mω(E) ≡ lim Re mω(E + iη),
η0
Im mω(E) ≡ lim Im mω(E + iη),
η0

E ∈ R, whenever the limits exist.

Choosing ω to be the standard semicircular law ρsc, the Stieltjes transform mρsc ≡ msc can be computed explicitly, and one checks that msc satisfies the relation

msc(z)

=

−1 msc(z) +

, z

Im msc(z) ≥ 0

z ∈ C+ .

The deformed semicircle law is conveniently defined through its Stieltjes trans-

form. Let ν be the limiting probability measure of Assumption 2.2. Then it is well

known [46] that the functional equation

(2.9)

mfc(z) =

dν(v) ,
v − z − mfc(z)

Im mfc(z) ≥ 0

z ∈ C+ ,

has a unique solution, also denoted by mfc, that satisfies, for all E ∈ R, lim supη 0 Im mfc(E + iη) < ∞. Indeed, from (2.9), we obtain that

(2.10)

|v

−

dν(v) z − mfc(z)|2

=

Im mfc(z) Im mfc(z) +

η

≤

1

z ∈ C+ ,

thus |mfc(z)| ≤ 1, for all z ∈ C+. The deformed semicircle law, denoted by ρfc, is then defined through its density

1

ρfc(E)

:=

lim
η0

π

Im mfc(E

+

iη)

(E ∈ R).

The measure ρfc has been studied in detail in [6]. For example, it was shown there that the density ρfc is an analytic function inside the support of the measure.
The measure ρfc is also referred to as the additive free convolution of the semicircular law and the measure ν. More generally, the additive free convolution of
two (probability) measures ω1 and ω2, usually denoted by ω1 ω2, is defined as the distribution of the sum of two freely independent noncommutative random
variables, having distributions ω1, ω2, respectively; we refer, for example, to [1, 59] for reviews. Similar to (2.9), the free convolution measure ω1 ω2 can be described in terms of a set of functional equations for the Stieltjes transforms; see
[5, 16].
Our second assumption on ν guarantees (see Lemma 3.5 below) that ρfc is supported on a single interval and that ρfc has a square root behavior at the two endpoints of its support. Sufficient conditions for this behavior have been presented

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2357

in [52]. The assumptions below also rule out the possibility that the matrix H has “outliers” in the limit of large N .

ASSUMPTION 2.3. Let Iν be the smallest interval such that supp ν ⊆ Iν . Then there exists > 0 such that

(2.11)

inf
x∈Iν

dν(v) (v − x)2 ≥ 1 + .

Similarly, let Iν be the smallest interval such that supp ν ⊆ Iν . Then:

(1) for random (vi), there is a constant t > 0, such that

(2.12)

dν(v)

P inf
x∈Iν

(v − x)2 ≥ 1 +

≥ 1 − N −t,

for N sufficiently large;

(2) for deterministic (vi),

(2.13)

inf
x∈Iν

dν(v) (v − x)2

≥

1

+

,

for N sufficiently large.

We give two examples for which (2.11) is satisfied:

(1)

Choosing

ν

=

1 2

(δ−a

+ δa),

a

≥ 0,

we

have

Iν

= [−a, a].

For

a

< 1,

one

checks that there is a = (a) such that (2.11) is satisfied and that the deformed

semicircle law is supported on a single interval with a square root type behavior at

the edges. However, for a > 1, the deformed semicircle law is supported on two

disjoint intervals; for further details, see [2, 8, 9].

(2) Let ν be a centered Jacobi measure of the form

(2.14)

ν(v) = Z−1(v − 1)a(1 − v)bd(v)1[−1,1](v),

where d ∈ C1([−1, 1]), d(v) > 0, −1 < a, b < ∞ and Z, a normalization constant.

Then for a, b < 1, there is > 0 such that (2.11) is satisfied with Iν = [−1, 1]. However, if a > 1 or b > 1, then (2.3) may not be satisfied. In this setting the

deformed semicircle law is still supported on a single interval; however, the square

root behavior at the edge may fail. We refer to [39, 40] for a detailed discussion.

LEMMA 2.4. Let ν satisfy (2.11) for some > 0. Then there are L−, L+, with L− ≤ −2, 2 ≤ L+, such that supp ρfc = [L−, L+]. Moreover, ρfc has a strictly
positive density in (L−, L+).

Lemma 2.4 follows directly from Lemma 3.5 below.

2.3. Results on bulk universality.

Recall that we denote by

N H ,n

the

n-point

correlation function of H = V + W , where V is either a real deterministic or real

random diagonal matrix. We denote by

N G,n

the

n-point

correlation

function

of

the

GUE, respectively, the GOE.

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LEE, SCHNELLI, STETLER AND YAU

A function O : Rn → R is called an n-particle observable if O is symmetric,
smooth and compactly supported. Recall from Lemma 2.4 that we denote by L± the endpoints of the support of the measure ρfc. For deterministic V we have the following result.

THEOREM 2.5. Let W be a complex Hermitian or a real symmetric Wigner
matrix satisfying the assumptions in Definition 2.1. Let V be a deterministic real diagonal matrix satisfying Assumptions 2.2 and 2.3. Set H = V + W . Let E, E be two energies satisfying E ∈ (L−, L+), E ∈ (−2, 2). Fix n ∈ N, and let O be an n-particle observable. Let δ > 0 be arbitrary, and choose b ≡ bN such that N −δ ≥ bN ≥ N −1+δ. Then

lim
N →∞

dα1 · · · dαnO(α1, . . . , αn)
Rn

(2.15)

×

1 2b

E+b dx E−b [ρfc(E)]n

N H ,n

x + α1 , . . . , x + αn

ρfc(E)N

ρfc(E)N

−

1 [ρsc(E

)]n

N G,n

E + α1 , . . . , E + αn

ρsc(E )N

ρsc(E )N

= 0,

where ρfc denotes the density of the deformed semicircle law and ρsc denotes the

density of the standard semicircle law. Here,

N G,n

denotes

the

n-point

correlation

function of the GUE in case W is a complex Hermitian Wigner matrix, respectively,

the n-point correlation function of the GOE in case W is a real symmetric Wigner

matrix.

For random V we have the following result.

THEOREM 2.6. Let W be a complex Hermitian or a real symmetric Wigner matrix satisfying the assumptions in Definition 2.1. Let V be a random real diagonal matrix whose entries are i.i.d. random variables that are independent of W and satisfy Assumptions 2.2 and 2.3. Set H = V +W . Let E, E be two energies satisfying E ∈ (L−, L+), E ∈ (−2, 2). Fix n ∈ N, and let O be an n-particle observable. Let δ > 0 be arbitrary, and choose b ≡ bN such that N −δ ≥ bN ≥ N −1/2+δ. Then

lim
N →∞

dα1 · · · dαnO(α1, . . . , αn)
Rn

(2.16)

×

1 2b

E+b dx E−b [ρfc(E)]n

N H ,n

x + α1 , . . . , x + αn

ρfc(E)N

ρfc(E)N

−

1 [ρsc(E

)]n

N G,n

E + α1 , . . . , E + αn

ρsc(E )N

ρsc(E )N

= 0,

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2359

where ρfc denotes the density of the deformed semicircle law and ρsc denotes the

density of the standard semicircle law. Here,

N G,n

denotes

the

n-point

correlation

function of the GUE in case W is a complex Hermitian Wigner matrix, respectively,

the n-point correlation function of the GOE in case W is a real symmetric Wigner

matrix.

REMARK 2.7. Theorem 2.5 and Theorem 2.6 show that the averaged local

correlation functions of H = V + W are universal in the limit of large N in the

sense that they are independent of the diagonal matrix V and also independent of

the precise distribution of the entries of W . Both theorems hold for real symmet-

ric and complex Hermitian matrices. For the former choice,

N G,n

stands

for

the

n-point correlation functions of the GOE. For the latter choice,

N G,n

stands

for

the

n-point correlation functions of the GUE.

Note that we can choose bN of order N −1+δ, δ > 0, for deterministic V in

Theorem 2.5, while we have to choose bN of order N −1/2+δ, δ > 0, for random V

in Theorem 2.6. The latter condition is technical and not optimal. It is related to

our next comment.

For random V with (vi) i.i.d. bounded random variables, the eigenvalues of H fluctuate on scale N −1/2 in the bulk [39]. Yet, under the assumptions of Theo-

rem 2.6, the eigenvalue gaps remain rigid over small scales so that the universal-

ity of local correlation functions, a statement about the eigenvalue gaps, is unaf-

fected by these mesoscopic fluctuations. We thus expect Theorem 2.6 to hold with bN N −1. Relying on explicit integration formulas in the complex Hermitian

setting, we suppose that the averaging over an energy window can be dropped; cf.

the results for the deformed GUE in [51].

REMARK 2.8. The main ingredient of our proofs of Theorem 2.5 and Theorem 2.6 is an entropy estimate; see Proposition 5.3. Once such an estimate is obtained, the method in [31] also implies the single gap universality in the sense that the distribution of any single gap in the bulk is the same (up to a scaling) as the one from the corresponding Gaussian case. More precisely, fix α > 0, and let k ∈ N be such that αN ≤ k ≤ (1 − α)N . Let O be an n-particle observable. Then there are χ > 0 and C such that

EH O (Nρfc,k)(λk − λk+1), (Nρfc,k)(λk − λk+2), . . . , (Nρfc,k)(λk − λk+n) − EμG O (Nρsc,k)(λk − λk+1), (Nρsc,k)(λk − λk+2), . . . ,

≤ CN −χ ,

(Nρsc,k)(λk − λk+n)

for N sufficiently large, where μG is the standard GOE or GUE ensemble, depending on the symmetry class of H . Here ρfc,k stands for the density of the measure

2360

LEE, SCHNELLI, STETLER AND YAU

ρfc at the classical location, γk, of the kth eigenvalue defined through

(2.17)

γk −∞

ρfc(x)

dx

=

k

− 1/2 .
N

Similarly, ρsc,k stands for the density of the standard semicircle law ρsc at the classical location of the kth eigenvalue of the Gaussian ensembles.

REMARK 2.9. To conclude, we mention two extensions of the above results. In Theorem 2.6 we may relax the assumption that (vi) are independent among themselves: our results can be extended to dependent random variables provided that (vi) satisfy (2.6), (2.11) and (2.12) for some constants α0, , t > 0, and provided that (vi) are independent of (wij ). In such a setting the required lower bound on bN depends on α0.
The assumption that V is diagonal can be relaxed by assuming in turn that W belongs to the GUE/GOE. Then using the invariance of W , we can diagonalize V and apply our approach for diagonal potentials. For W a Wigner matrix and V a nondiagonal matrix, we expect that similar results hold by slowly changing W to a GUE/GOE. This, however, involves many more technical steps.

2.4. Results on edge universality. In this subsection, we show that our model also satisfies the edge universality. Edge universality states that the statistics of the extremal eigenvalues of many random matrix ensembles are universal: let λN denote the largest eigenvalue of a Wigner matrix W . The limiting distribution of λN was identified for the Gaussian ensembles by Tracy and Widom [57, 58]. They proved that

(2.18)

lim P
N →∞

N 2/3(λN

− 2)

≤

s

= Fβ (s)

β ∈ {1, 2, 4} ,

s ∈ R, where the Tracy–Widom distribution functions Fβ are described by Painlevé equations. The edge universality can also be extended to the k largest eigenvalues, where the joint distribution of the k largest eigenvalues can be written in terms of the Airy kernel, as first shown for the GUE/GOE in [34]. These results also hold for the k smallest eigenvalues.
Edge universality for Wigner matrices was first proved in [54] (see also [53]) for real symmetric and complex Hermitian ensembles with symmetric distributions. The symmetry assumption on the entries’ distribution was partially removed in [47, 48]. Edge universality was proved in [55] under the condition that the distribution of the matrix elements has subexponential decay, and its first three moments match those of the Gaussian distribution; that is, the third moment of the entries vanish. The vanishing third moment condition was removed in [33]. Finally, edge universality for generalized Wigner matrices was proved only recently in [12].
Edge universality for the deformed GUE was obtained for the special case when V has two eigenvalues ±a, each with equal multiplicity, via a Riemann– Hilbert approach in [2, 8]. For general V , the joint distribution of the eigenvalues

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2361

of the deformed GUE can be expressed explicitly by the Brezin–Hikami/Johansson formula that may be used to prove the edge universality various choices and ranges of V ; see [14, 36, 51].
Our result on the edge universality for real symmetric and complex Hermitian deformed Wigner matrices is as follows.

THEOREM 2.10. Let W be a complex Hermitian or a real symmetric Wigner matrix satisfying the assumptions in Definition 2.1. Let V be either a random real diagonal matrix whose entries are i.i.d. random variables that are independent of W , or a deterministic real diagonal matrix. Assume that V satisfies Assumptions 2.2 and 2.3. Set H = V + W .
Then there are κ > 0, χ > 0, c0 > 0 such that the following result holds for any fixed n ∈ N. For any n-particle observable O and for ⊂ [[1, N κ]], respectively,
⊂ [[N − N κ, N ]], with | | = n, we have

(2.19)

EH O c0N 2/3j 1/3(λj − γj ) j∈ ≤ CO N −χ ,

− EμG O N 2/3j 1/3(λj − γj ) j∈

for N sufficiently large, for some constant CO (depending on O), where μG is the standard GUE/GOE, depending on the symmetry class of W . Here, the constant c0 is a scaling factor so that the eigenvalue density at the edge of H can be compared
with the Gaussian case. It only depends on ν. Further, γj , γj denote here the classical locations of the j th eigenvalue with respect to the measure ρfc introduced in (3.8) below, respectively, with respect to the standard semicircle law ρsc.

Theorem 2.10 shows that the local statistics of the k largest, respectively, smallest, eigenvalues of our model are given by the Tracy–Widom–Airy statistics.
The measure fc depends solely on the empirical eigenvalue distribution, ν, of V , and so do the classical locations (γk). The scaling factor c0 in (2.19) may be computed explicitly [51].
Theorem 2.10 is proved in a similar way to Theorems 2.5 and 2.6. Using the Dirichlet form bound obtained in Proposition 5.3 below, we invoke the edge universality result for localized β-ensembles, Theorem 3.3 of [12], and follow the same strategy as for the bulk universality. The proof of Theorem 2.10 is given in Section 9.
To conclude, we mention that Theorem 2.10 has recently been proved in [41] using a completely different approach based on the Green function comparison theorem; see, for example, [32] for earlier ideas of using the Green function comparison for edge universality.

2.5. Notation and conventions. In this subsection, we introduce some more

notation and conventions used throughout the paper. For high probability estimates

we use two parameters ξ ≡ ξN and ϕ ≡ ϕN : we let

(2.20)

a0 < ξ ≤ A0 log log N, ϕ = (log N )C1,

2362

LEE, SCHNELLI, STETLER AND YAU

for some constants a0 > 2, A0 ≥ 10, C1 > 1.

DEFINITION 2.11. We say an event has (ξ, υ)-high probability if

P c ≤ e−υ(log N)ξ

(υ > 0),

for N sufficiently large. We say an event has ς -exponentially high probability if

P c ≤ e−Nς

(ς > 0),

for N sufficiently large. Similarly, for a given event 0 we say an event holds with (ξ, υ)-high probability, respectively, ς -exponentially high probability, on 0, if

P c ∩ 0 ≤ e−υ(log N)ξ

(υ > 0),

P c ∩ 0 ≤ e−Nς

(ς > 0),

respectively, for N sufficiently large.

For brevity, we occasionally say an event holds with exponentially high probability, when we mean ς -exponentially high probability. We do not keep track of the explicit value of υ or ς in the following, allowing υ and ς to decrease from line to line such that υ, ς > 0.
We use the symbols O(·) and o(·) for the standard big-O and little-o notation. The notation O, o, , , refers to the limit N → ∞, if not indicated otherwise. Here a b means a = o(b). We use c and C to denote positive constants that do not depend on N . Their value may change from line to line. We write a ∼ b if there is C ≥ 1 such that C−1|b| ≤ |a| ≤ C|b|, and occasionally we write for N dependent quantities aN bN if there exist constants C, c > 0 such that |aN | ≤ C(ϕN )cξ |bN |.
Finally, we abbreviate
(i) N
(·) ≡ (·),
j j =1 j =i
and we use double brackets to denote index sets, that is,
[[n1, n2]] := [n1, n2] ∩ Z,
for n1, n2 ∈ R.

3. Local law and rigidity estimates. Recall the constant > 0 in Assumption 2.3. Set := /10. In this section we consider the family of interpolating random matrices
(3.1) H ϑ := ϑV + W, ϑ ∈ := 0, 1 + ,

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2363

where V and W are chosen to satisfy Assumptions 2.2 and 2.3, respectively,
the assumptions in Definition 2.1. Here ϑ has the interpretation of a possibly
N -dependent positive “coupling parameter.” We define the resolvent or Green function, Gϑ (z), and the averaged Green func-
tion, mϑ (z), of H ϑ by

(3.2)

Gϑ (z) =

Gϑij (z)

:=

ϑV

1 +W

−

, z

mϑN (z)

:=

1 N

Tr Gϑ (z),

z ∈ C+. Frequently, we abbreviate Gϑ ≡ Gϑ (z), mϑN ≡ mϑN (z), etc. To conveniently cope with the cases when (vi) are random, respectively, de-
terministic, we introduce an event on which the random variables (vi) exhibit
“typical” behavior. Recall that we denote by mν and mν the Stieltjes transforms
of ν, respectively, ν.

DEFINITION 3.1. Let ≡ (N ) be an event on which the following holds:

(1) There is a constant α0 > 0 such that, for any fixed compact set D ⊂ C+ (independent of N ) with dist(D, supp ν) > 0, there is C such that

(3.3)

mν (z) − mν(z) ≤ CN −α0,

for N sufficiently large. (2) Recall the constant > 0 in Assumption 2.3. We have

(3.4)

dν(v)

inf
x∈Iν

(v − x)2 ≥ 1 + ,

dν

inf
x∈Iν

(v − x)2 ≥ 1 + ,

for N sufficiently large.

In case (vi) are deterministic, has full probability for N sufficiently large by
the Assumptions in 2.2. Similar to the definition of mfc, we define mϑfc and mϑfc as the solutions to the
equations

(3.5) and

mϑfc(z) =

dν(v) ϑv − z − mϑfc(z) ,

Im mϑfc(z) ≥ 0

z ∈ C+

(3.6)

mϑfc(z) =

dν(v) ϑv − z − mϑfc(z) ,

Im mϑfc(z) ≥ 0,

z ∈ C+ ,

respectively. Following the discussion of Section 2.2, mϑfc and mϑfc define two probability measures ρfϑc and ρfϑc through the densities

(3.7)

ρfϑc(E)

:=

lim
η0

1 π

Im mϑfc(E

+

iη)

(E ∈ R)

2364

LEE, SCHNELLI, STETLER AND YAU

and

(3.8)

ρfϑc(E)

:=

lim
η0

1 π

Im mϑfc(E

+

iη)

(E ∈ R);

cf. (2.7). More precisely, we have the following result which follows directly from the proofs of Lemmas 3.5 and 3.6 below. Recall the definition of in (3.1).

LEMMA 3.2. Let ν and ν satisfy the Assumptions 2.2 and 2.3. Then, for any
ϑ ∈ and N ∈ N, equations (3.5) and (3.6) define, through the inversion formulas in (3.7) and (3.8), absolutely continuous measures ρfϑc and ρfϑc. Moreover, the measure ρfϑc is supported on a single interval with strictly positive density inside this interval. The same holds true on for the measures ρfϑc, for N sufficiently large.

Note that if (vi) are random, then so are mϑfc, respectively, ρfϑc. As noted above, we use the symbol to denote quantities that depend on the empirical distribution ν of the (vi), while we drop this symbol for quantities depending on the limiting distribution ν of (vi).
We denote by Lϑ±, respectively, Lϑ±, the endpoints of the support of ρfϑc, respectively, ρfϑc. Let E0 ≥ 1 + max{|L1−|, L1+}, and define the domain
(3.9) DL := z = E + iη ∈ C : |E| ≤ E0, (ϕN )L ≤ N η ≤ 3N ,
with L ≡ L(N), such that L ≥ 12ξ ; see (2.20). The following theorem is the main result of this section.

THEOREM 3.3 (Strong local deformed semicircle law). Let H ϑ = ϑV + W , ϑ ∈ [see (3.1)], where W is a real symmetric or complex Hermitian Wigner matrix satisfying the assumptions in Definition 2.1 and V is a deterministic or random real diagonal matrix satisfying Assumptions 2.2 and 2.3. Let

(3.10)

ξ = A0 + o(1) log log N. 2

Then there are constants υ > 0 and c1, depending on the constants E0 in (3.9), α0 in (3.3), A0, a0, C1 in (2.20), θ , C0 in (2.3) and the measure ν such that the following holds for L ≥ 40ξ . For any z ∈ DL and any ϑ ∈ , we have

(3.11)

mϑN (z) − mϑfc(z)

≤

(ϕN

)c1ξ

1 Nη

,

with (ξ, υ)-high probability on . Moreover, we have, for any z ∈ DL, any ϑ ∈

and any i, j ∈ [[1, N]],

(3.12)

Gϑij (z) − δij giϑ (z) ≤ (ϕN )c1ξ

Im mϑfc(z) + 1 , Nη Nη

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2365

with (ξ, υ)-high probability on , where we have set

(3.13)

giϑ (z)

:=

ϑ vi

−

1 z−

mϑfc(z) .

The study of local laws for Wigner matrices was initiated in [25–27]. For more

recent results, we refer to [23]. For deformed Wigner matrices with random poten-
tial, a local law was obtained in [39]. Denote by λϑ = (λϑ1 , λϑ2 , . . . , λϑN ) the eigenvalues of the random matrix H ϑ =
ϑV + W arranged in ascending order. We define the classical location, γiϑ , of the eigenvalue λϑi by

(3.14)

γiϑ −∞

ρfϑc(x)

dx

=

i

− (1/2) N

(1 ≤ i ≤ N).

Note that (γiϑ ) are random in case (vi) are too. We have the following rigidity result on the eigenvalue locations of H ϑ :

COROLLARY 3.4. Let H ϑ = ϑV + W , ϑ ∈ , where W is a real symmetric or complex Hermitian Wigner matrix satisfying the assumptions in Definition 2.1, and V is a deterministic or random real diagonal matrix satisfying Assumptions 2.2 and 2.3. Let ξ satisfy (3.10). Then there are constants υ > 0 and c1, c2, depending on the constants E0 in (3.9), α0 in (3.3), A0, a0, C1 in (2.20), θ , C0 in (2.3) and the measure ν, such that

(3.15)

λϑi − γiϑ

≤

(ϕN

)c1ξ

N

1 2/3αˇ i1/3

(1 ≤ i ≤ N),

(3.16)

N i=1

λϑi

− γiϑ

2

≤ (ϕN )c2ξ

1 ,
N

with (ξ, υ)-high probability on , for all ϑ ∈ , where we have abbreviated αˇ i := min{i, N − i + 1}.

In the rest of this section we sum up the proofs of Theorem 3.3 and Corollary 3.4.

3.1. Properties of mϑfc and mϑfc. In this subsection, we discuss properties of the Stieltjes transforms mϑfc and mϑfc. We first derive the desired properties for mϑfc (Lemma 3.5 and Corollary A.2 in the Appendix) and then show in a second step that mϑfc is a good approximation to mϑfc so that mϑfc also shares these properties; see Lemma 3.6.
For E0 as in (3.17), we define the domain, D , of the spectral parameter z by

(3.17)

D := z = E + iη : E ∈ [−E0, E0], η ∈ (0, 3] .

2366

LEE, SCHNELLI, STETLER AND YAU

The next lemma, whose proof is postponed to the Appendix, gives a qualitative description of the deformed semicircle law ρfϑc and its Stieltjes transform mϑfc.

LEMMA 3.5. Let ν satisfy Assumption 2.3, for some > 0. Then the fol-

lowing holds true for any ϑ ∈ . There are Lϑ−, Lϑ+ ∈ R, with Lϑ− < 0 < Lϑ+, such that supp ρfϑc = [Lϑ−, Lϑ+], and there exists a constant C > 1 such that, for all ϑ∈ ,

(3.18)

C−1√κE

≤

ρfϑc(E)

≤

√ C κE

E ∈ Lϑ−, Lϑ+ ,

where κE denotes the distance of E to the endpoints of the support of ρfϑc, that is,

(3.19)

κE := min E − Lϑ− , E − Lϑ+ .

The Stieltjes transform, mϑfc, of ρfϑc has the following properties:

(1) for all z = E + iη ∈ D ,

(3.20)

⎧ ⎪⎨

√ κ

+

η,

Im mϑfc(z) ∼ ⎪⎩ √κη+ η ,

E ∈ Lϑ−, Lϑ+ , E ∈ Lϑ−, Lϑ+ c;

(2) there exists a constant C > 1 such that for all z ∈ D and all x ∈ Iν,

(3.21)

C−1 ≤ ϑx − z − mϑfc(z) ≤ C.

Moreover, the constants in (3.18), (3.20) and (3.21) can be chosen uniformly in ϑ∈ .

Next, we argue that mϑfc behaves qualitatively in the same way as mϑfc on for N sufficiently large. Lemma 3.6 below is proven in the Appendix.

LEMMA 3.6. Let ν satisfy Assumptions 2.2 and 2.3, for some > 0. Then
the following holds for all ϑ ∈ and all sufficiently large N on . There are Lϑ−, Lϑ+ ∈ R, with Lϑ− < 0 < Lϑ+, such that supp ρfϑc = [Lϑ−, Lϑ+]. Let κE := min{|E − Lϑ−|, |E − Lϑ+|}. Then (3.18), (3.20) and (3.21) of Lemma 3.5, hold true on , for N sufficiently large, with mϑfc replaced by mϑfc, ρfϑc replaced by ρfϑc, etc. Moreover, the constants in these inequalities can be chosen uniformly in ϑ ∈
and N , for N sufficiently large.
Further, there is c > 0 such that for all z ∈ D we have

(3.22)

mϑfc(z) − mϑfc(z) ≤ N −cα0/2

Lϑ± − Lϑ± ≤ N −cα0 ,

on for N sufficiently large and all ϑ ∈ .

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2367

3.2. Proof of Theorem 3.3 and Corollary 3.4. The proof of Theorem 3.3 fol-
lows closely the proof of Theorem 2.10 in [39]. The difference between Theo-
rem 3.3 of the present paper and Theorem 2.10 in [39] is that we presently con-
dition on the diagonal entries (vi); that is, we consider the entries of V as fixed. Accordingly, we compare [on the event of typical (vi)] the averaged Green function mϑ with mϑfc [see (3.6)] instead of mϑfc; see (3.5). For consistency, we momentarily drop the ϑ dependence form our notation. To establish Theorem 3.3,
we first derive a weak local deformed semicircle law (see Theorem 4.1 in [39]) by
following the proof in [39]. Using the Lemma 3.5, Lemma 3.6 and the results in
the Appendix, it is then straightforward to obtain the following result.

LEMMA 3.7. Under the assumption of Theorem 3.3, there are c1 and υ > 0 such that

mN (z) − mfc(z)

≤

(ϕN

)c1ξ

(N

1 η)1/3

,

Gij (z)

≤

(ϕN

)c1ξ

√1 N

η

,

with (ξ, υ)-high probability on , uniformly in z ∈ DL and ϑ ∈ .

To prove Theorem 3.3 we follow mutatis mutandis the proof of Theorem 4.1
in [39]. But we note that in the corresponding equation to (5.25) in [39], we may set λ = 0 in the error term, at the cost of replacing mfc by mfc. In the subsequent analysis, we can simply set λ = 0 in the error terms. In this way, one establishes the proof of Theorem 3.3. Similarly, Corollary 3.4 can be proven in the same way as is Theorem 2.21 in [39]. It suffices to set λ = 0 in the analysis in [39]. We leave the details aside.

4. Reference β-ensemble.

4.1. Definition of β-ensemble and known results. We first recall the notion of β-ensembles. Let N ∈ N, and let (N) ⊂ RN denote the set

(4.1)

(N) := x = (x1, x2, . . . , xN ) : x1 ≤ x2 ≤ · · · ≤ xN .

Consider the probability distribution, μU ≡ μNU , on (N) given by

(4.2)

μNU

(dx)

:=

1 ZUN

e−β N H(x)

dx,

dx := 1 x ∈ (N) dx1 dx2 · · · dxN ,

where β > 0,

(4.3)

N1 H(x) :=
i=1 2

U (xi)

+

xi2 2

1 − N 1≤i<j≤N log(xj − xi )

and ZUN ≡ ZUN (β) is a normalization. Here U is a potential, that is, a real-valued, sufficiently regular function on R. In the following, we often omit the parameters

2368

LEE, SCHNELLI, STETLER AND YAU

N and β from the notation. We use PμU and EμU to denote the probability and the
expectation with respect to μU . We view μU as a Gibbs measure of N particles on R with a logarithmic interaction, where the parameter β > 0 may be interpreted as the inverse temperature. (For the results in the present paper, we choose β = 2 in case W is complex Hermitian Wigner matrix and β = 1 in case W is a real
symmetric Wigner matrix.) We refer to the variables (xi) as particles or points, and we call the system a log-gas or a β-ensemble. We assume that the potential U is a C4 function on R such that its second derivative is bounded below; that is, we
have

(4.4)

inf U
x∈R

(x) ≥ −2CU ,

for some constant CU ≥ 0, and we further assume that

(4.5)

U (x) + x2 > (2 + ε) log 1 + |x|

(x ∈ R),

2

for some ε > 0, for large enough |x|. It is well known (see, e.g., [13]) that under these conditions the measure is normalizable, ZUN < ∞. Moreover, the averaged density of the empirical spectral measure, ρUN , defined as

(4.6)

ρUN

:= EμU

1 N

N
δxi ,
i=1

converges weakly in the limit N → ∞ to a continuous function ρU , the equilibrium density, which is of compact support. It is well known that ρU can be obtained as the unique solution to the variational problem

inf x2 + U (x) dρ(x) − log |x − y| dρ(x) dρ(y) :

(4.7)

R2

R

ρ is a probability measure

and that the equilibrium density ρ = ρU satisfies

(4.8)

U (x) + x = −2− ρ(y) dy R y−x

(x ∈ supp ρU ).

In fact, (4.8) holds if and only if x ∈ supp ρU . We will assume in addition that the minimizer ρU is supported on a single interval [A−, A+] and that U is “regular” in the sense of [38]; that is, the equilibrium density of U is positive on (A−, A+) and vanishes like a square root at each of the endpoints of [A−, A+]. Viewing the points x = (xi) as points or particles on R, we define the classical location of the
kth particle, γk, under the β-ensemble μU by

(4.9)

γk k − (1/2)

ρU (x) dx =
−∞

N

.

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2369

For a detailed discussion of general β-ensemble we refer, for example, to [1, 12]. For U ≡ 0, we write μG ≡ μNG instead of μ0, since μ0 is the equilibrium mea-
sure for the GUE (β = 2), respectively, the GOE (β = 1). More precisely, setting

(4.10)

HG(x)

:=

N i=1

1 4

xi2

−

1 N

1≤i<j ≤N

log(xj

−

xi ),

the GUE, respectively, GOE, distribution on (N) are given by

(4.11)

μNG (dx)

=

1 ZGN

e−β N HG (x)

dx,

where ZGN ≡ ZGN (β) is a normalization, and we either choose β = 2 or β = 1. We are interested in the n-point correlation functions defined by

(4.12)

UN,n(x1, . . . , xn) = RN−n μ#U (x) dxn+1 · · · dxN ,

where μ#U is the symmetrized version of μU given in (4.2) but defined on RN instead of the simplex (N),

(4.13)

μ#U (dx) =

1 N ! μU

dx(σ )

,

dx = dx1 · · · dxN ,

where x(σ ) = (xσ (1), . . . , xσ (N)), with xσ (1) < · · · < xσ (N). The following universality result is proven in [12].

THEOREM 4.1 (Bulk universality for β-ensembles, Theorem 2.1 in [12]). Let

U be a C4 regular potential with equilibrium density supported on a single in-

terval [A−, A+] that satisfies (4.4) and (4.5). Then the following result holds. For

any

fixed

β

> 0,

E

∈ (A−, A+),

|E

| < 2,

n ∈ N,

0<δ

≤

1 2

and

any

n-particle

observable O, we have with b := N −1+δ,

lim
N →∞

dα1 · · · dαnO(α1, . . . , αn)
Rn

×

E+b dx 1 E−b 2b [ρU (E)]n

N U,n

x + α1 , . . . , x + αn

NρU (E)

NρU (E)

−

1 [ρsc(E

)]n

N G,n

E + α1 , . . . , E + αn

Nρsc(E )

Nρsc(E )

= 0.

Here, ρsc denotes the density of the semicircle law, and

N G,n

is

the

n-point

the

correlation function of the Gaussian β-ensemble, that is, with U ≡ 0.

Theorem 4.1 was first proved in [11] under the assumption that U is analytic, a hypothesis that was only required for proving rigidity. The analyticity assumption has been removed in [12]. Recently, alternative proofs of bulk universality for

2370

LEE, SCHNELLI, STETLER AND YAU

β-ensembles with general β > 0, that is, results similar to Theorem 4.1, have been obtained in [50] and [4]. In the present paper, we will not use Theorem 4.1; it is stated here for completeness.
To conclude this subsection, we recall an important tool in the study of β-ensembles, the “first order loop” equation. In the notation above it reads (in the limit N → ∞)

(4.14)

mU (z)2 =

x + U (x) x − z ρU (x) dx

z ∈ C+ ,

where mU denotes the Stieltjes transform of the equilibrium measure ρU , that is,

mU (z) ≡ mρU (z) =

ρU (x) dx x−z

z ∈ C+ .

The loop equation (4.14) can be obtained by a change of variables in (4.2) (see [35]) or by integration by parts; see [49].

4.2. Time-dependent modified β-ensemble. In this subsection, we introduce a
modified β-ensemble by specifying potentials U and U that depend, among other things, on a parameter t ≥ 0 which has the interpretation of a time. The potential U also depends on N , the size of our original matrix H = V + W , yet the N dependence is only through the fixed random variables (vi). Recall that we have defined mϑfc, respectively, mϑfc, as the solutions to the equations

(4.15)

mϑfc(z) =

dν(v) ϑvi − z − mϑfc(z) ,

mϑfc(z) =

dν(v) ϑv − z − mϑfc(z) ,

z ∈ C+, subject to the conditions Im mϑfc(z), Im mϑfc(z) ≥ 0, for Im z > 0. Recall from (3.1) that we denote = [0, 1 + ], = /10. We then fix some t0 ≥ 0 such that et0/2 ∈ and let

(4.16)

ϑ ≡ ϑ(t) := e−(t−t0)/2 (t ≥ 0).

In the following we consider t ≥ 0 as time, and we henceforth abbreviate mϑfc(t)(z) ≡ mfc(t, z), etc. Equation (4.15) defines time dependent measures ρfc(t), ρfc(t), respectively, whose densities at the point x ∈ R are denoted by ρfc(t, x), respectively, ρfc(t, x).
We denote by U (t, x), U (n)(t, x) the first, respectively, the nth derivative of
U (t, x) with respect to x, and we use the same notation for U . We define U and U
(up to finite additive constants that enter the formalism only in normalizations) through their derivatives U and U . For t ≥ 0, we set

(4.17)

U

(t, x)

+

x

:=

−2−
R

ρfc(t, y) y−x

dy,

for x ∈ supp ρfc(t), respectively,

(4.18)

U (t, x) + x := −2− ρfc(t, y) dy, R y−x

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2371

for x ∈ supp ρfc(t). Outside the support of the measures ρfc(t) and ρfc(t), we define U and U as C3 extensions such that they are “regular” potentials satisfying (4.4) and (4.5) for all t ≥ 0. The definitions of such potentials are obviously not unique. One possible construction is outlined in the Appendix in the form of the proof of the next lemma.

LEMMA 4.2. There exist potentials U , U : R+ × R → R, (t, x) → U (t, x), U (t, x) such that for n ∈ [[1, 4]], U (n)(t, x), U (n)(t, x), ∂t U (n)(t, x), ∂t U (n)(t, x) are continuous functions of x ∈ R and t ∈ R+, which can be uniformly bounded in x on compact sets, uniformly in t ∈ R+ and sufficiently large N . Moreover the
following holds for all t ≥ 0 on for N sufficiently large:

(1) U (t, x) and U (t, x) satisfy (4.17) and (4.18) for x ∈ supp ρfc(t), respectively, x ∈ supp ρfc(t). For x ∈/ supp ρfc(t), respectively, x ∈/ supp ρfc(t), we have

U (t, x) + x > 2 Re mfc(t, x) , U (t, x) + x > 2 Re mfc(t, x) .

(2) There is a constant c > 0 such that for all x ∈ R and all t ≥ 0, we have

(4.19)

U (t, x) − U (t, x) ≤ N −cα0/2,

where α0 > 0 is the constant in (3.3). (3) The potentials U and U satisfy (4.4) and (4.5). In particular, there is
CU ≥ 0 (independent of N ), such that

(4.20)

inf U
x ∈R,t ∈R+

(t, x) ≥ −2CU ,

inf U
x ∈R,t ∈R+

(t, x) ≥ −2CU .

Moreover, U and U are “regular”; see the paragraph below (4.8) for the definition of “regular” potential.

Below, we are mainly interested in β-ensembles determined by the potential U .
For ease of notation, we thus limit the discussion to U . For N ∈ N we define a measure on (N) by setting

(4.21)

ψt (x)μG(dx)

:=

1 Zψt

e−((β N )/2)

N i=1

U

(t

,xi

)

μG(dx)

x ∈ (N) ,

where Zψt ≡ Zψt (β) is a normalization, and we usually choose β = 1, 2. By Lemma 4.2, ψt μG is a well-defined β-ensemble, and from the discussion in Section 4.1 we further infer that the equilibrium density of ψt μG, that is, the unique measure solving the minimization problem in (4.7), is for any t ≥ 0, ρfc(t). Viewing ψt μG as a Gibbs measure of N (ordered) particles (xi) on the real line, we define the classical location of the ith particles, γi(t), as in (4.9), that is,

(4.22)

γi(t) i − (1/2)

ρfc(t, x) dx =
−∞

N

i ∈ [[1, N]] .

From [12] we have the following rigidity result.

2372

LEE, SCHNELLI, STETLER AND YAU

PROPOSITION 4.3. Let U (t, ·), with t ≥ 0 and N ∈ N, be given by Lemma 4.2. Then the following holds on . For any δ > 0, there is ς > 0, such that for any t ≥ 0,

(4.23) Pψt μG xi − γi (t ) > N −(2/3)+δαˇ i−1/3 ≤ e−Nς

(1 ≤ i ≤ N),

for N sufficiently large, where Pψt μG stands for the probability under ψt μG conditioned on V . Here, αˇ i := min{i, N − i + 1}.

PROOF. The rigidity estimate (4.23) is taken from Theorem 2.4 of [12]. To achieve uniformity in t ≥ 0 and N sufficiently large, we note that estimate (4.23) depends on the potential mainly through the convexity bounds (4.4) and (4.5). Starting from the uniform bounds of Lemma 4.2, one checks that Proposition 4.3 holds uniformly in t and N large enough.

In the rest of this section, we derive equations of motion for the potential U (t, ·)
and the classical locations (γi(t)). To derive these equations we observe that the Stieltjes transform mfc(t, z) can be obtained from mfc(t = 0, z) as the solution to the following complex Burgers equation [46]:

(4.24)

∂t mfc(t, z)

=

1 2

∂z

mfc(t, z)

mfc(t, z) + z

z ∈ C+, t ≥ 0 .

This can be checked by differentiating (4.15). Combining the complex Burgers equation (4.24) and the loop equation (4.14) we obtain the following result.

LEMMA 4.4. Let N ∈ N. Assume that ν satisfies the Assumptions 2.2 and 2.3. Then the following holds on for N sufficiently large. For t ≥ 0, we have

(4.25) respectively,

∂t γi(t)

=

1 2

U

t, γi(t) ,

(4.26)

∂t

γi

(t )

=

−−
R

ρfc(t, y) y − γi(t)

dy

−

1 2

γi

(t )

i ∈ [[1, N]] .

Further, the potential U satisfies

(4.27)

∂t U (t, x)

=

−
R

U

(t, y)ρfc(t, y) y−x

dy

x ∈ supp ρfc(t) .

Moreover, there exist constants C, C such that the following bounds hold on

:

(4.28)

∂t γi(t) ≤ C,

∂t U (t, x) ≤ C ,

for all i ∈ [[1, N ]], uniformly in t ≥ 0, x ∈ supp ρfc(t) and N , for N sufficiently large.
Finally, U (t, ·) and (γi(t)), share the same properties.

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2373

PROOF. Combining (4.24) and (4.14), we find, for z ∈ C+, t ≥ 0,

1 ∂t mfc(t, z) = 2 ∂z −

v + U (t, v) v − z ρfc(t, v) dv + z

ρfc(t, v) v−z

dv

=

1 2 ∂z

−

U v

(t , −

v) z

ρfc(t

,

v)

dv

−

1

=

−

1 2

∂z

U (t, v) v − z ρfc(t, v) dv.

Hence, for Im z > 0, we get

∂t

mfc(t

,

z)

=

−

1 2

U (v

(t, v) − z)2

ρfc(t

,

v)

dv

=

−

1 2

(U (t, v)ρfc(t, v)) dv. (v − z)

Clearly U (t, v)ρfc(t, v) is a C3 function inside the support of ρfc(z) that has a square root behavior at the endpoints. Thus we obtain from the Stieltjes inversion
formula that

11

(4.29)

∂t ρfc(t, E)

=

π

lim
η0

Im

∂t

mfc(t

,

z)

=

− 2

U (t, E)ρfc(t, E)

,

for all E ∈ (L−(t), L+(t)), where L±(t) denote the endpoints of the support of ρfc (t ).
On the other hand, differentiating (4.22) with respect to time, we obtain

γi (t)
∂t ρfc(t, v) dv = −ρfc t, γi(t) ∂t γi(t).
−∞
Substituting from (4.29), we get

Hence

11

γi (t)

∂t γi(t) = 2 ρfc(t, γi(t)) −∞ dv Ufc(t, v)ρfc(t, v) .

11

∂t γi(t)

=

2

ρfc (t ,

U γi (t ))

t, γi(t) ρfc t, γi(t) ,

and (4.25) follows. Using that U satisfies (4.17), we can recast this last equation

as

∂t

γi

(t )

=

−−
R

ρfc (t , y − γi

y) (t )

dy

−

1 2

γi

(t ),

and we find (4.26). Equation (4.26) follows in a similar way by differentiat-

ing (4.17) with respect to time. By a similar computation we obtain (4.27). The

bound in (4.28) follows from Lemma 4.2.

Starting from the relations in (4.15), we derived via the time dependent potential U , an equation of motions for the classical locations (γi(t)). The points (γi(t))

2374

LEE, SCHNELLI, STETLER AND YAU

may also be viewed as the classical locations of the eigenvalues of a family of ran-
dom matrices which is parametrized by the times t0 and t. This is the subject of the next section.

5. Dyson Brownian motion: Evolution of the entropy.

5.1. Dyson Brownian motion. Let H0 = (hij,0) be the matrix H0 := et0/2V + W,

where V satisfies Assumptions 2.2 and 2.3, and W is real symmetric or complex Hermitian satisfying the assumptions in Definition 2.1. Here, t0 ≥ 0 is chosen such that ϑ = et0/2 ∈ [see (3.1)], and we consider ϑ as an a priori free “coupling parameter” that we fix in Section 8 below. Let B = (bij ) ≡ (bij,t ) be a real symmetric, respectively, a complex Hermitian, matrix whose entries are a collection
of independent, up to the symmetry constraint, real (complex) Brownian motions,
independent of (hij,0). More precisely, in case W is a complex Hermitian Wigner matrix, we choose the entries (bij,t ) to have variance t; in case W is a real symmetric Wigner matrix, we choose the off-diagonal entries of (bij,t ) to have variance t, while the diagonal entries are chosen to have variance 2t . Let Ht = (hij,t ) satisfy the stochastic differential equation

(5.1)

dhij

=

√dbij N

−

1 2 hij

dt

(t ≥ 0).

It is then easy to check that the distribution of Ht agrees with the distribution of the matrix

(5.2)

e−(t−t0)/2V + e−t/2W + 1 − e−t 1/2W ,

where W is, in case W is a complex Hermitian, a GUE matrix, independent of V

and W , respectively, a GOE matrix, independent of V and W , in case W is a real

symmetric Wigner matrix. The law of the eigenvalues of the matrix W is explicitly

given by (4.11) with β = 2, respectively, β = 1.

Denote by λ(t) = (λ1(t), λ2(t), . . . , λN (t)) the ordered eigenvalues of Ht . It is

well known that λ(t) satisfy the following stochastic differential equation:

(5.3)

√

dλi

=

√2 βN

dbi

+

− λi 2

+1 N

(i) j

1 λi − λj

dt

i ∈ [[1, N]] ,

where (bi) is a collection of real-valued, independent standard Brownian motions. If the matrix (bij ) in (5.1) is real symmetric, we have β = 1 in (5.3), respectively, β = 2, if (bij ) is complex Hermitian. The evolution of λ(t) is the celebrated Dyson Brownian motion [21].
For t ≥ 0, we denote by ft μG the distribution of λ(t). In particular, ft dμG ≡ ft (λ)μG(dλ) = 1. Note that ft μG depends on V through the initial condition

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2375

f0μG. In the following we always keep the (vi) fixed; that is, we condition on V . For simplicity, we omit this conditioning from our notation. The density ft is the solution of the equation

∂t ft = Lft (t ≥ 0),

where the generator L is defined via the Dirichlet form

(5.4) DμG(f ) = −

f

Lf

dμG

=

N i=1

1 βN

(∂if )2 dμG

Formally,

we

have

L

=

1 βN

− (∇HG) · ∇, that is,

(∂i ≡ ∂xi ).

(5.5)

L

=

N i=1

1 βN

∂i2

+

N i=1

11 − 2 λi + N

(i) 1 j λi − λj

∂i .

We remark that we use a different normalization in the definition of the Dirich-

let form DμG(f ) in (5.4) (and the generator L) than in earlier works, as in, for

example, [30], where the Dirichlet from is defined as

N1 i=1 2N

(∂if )2 dμG.

LEMMA 5.1 (Dyson Brownian motion). The equation ∂t ft = Lft , with initial data ft |t=0 = f0 has a unique solution on L1(μG) ≡ L1(RN , μG) for all t ≥ 0. Moreover, the domain (N) is invariant under the dynamics; that is, if f0 is supported in (N), then is ft for all t ≥ 0.

(Strictly speaking, the eigenvalue distribution of H0 may not allow a density f0, but for t > 0, Ht admits a density ft . Our proofs are not affected by this technicality.)
We refer, for example, to [1] for more details and proofs. To conclude, we record
one of the technical tools used in the next sections.

LEMMA 5.2. Denote by ft (λ)μG(dλ) the distribution of the eigenvalues of matrix (5.2) with t ≥ 0. Then, for any 0 < a < 1/2, we have

(5.6)

sup
t ≥0

1 N

N i=1

λi

− γi(t)

2ft (λ) dμ(λ) ≤

N −1−2a,

on for N sufficiently large, where (γi(t)) denote the classical locations with respect to the measure ρfc(t); that is, they are defined through the relation

(5.7)

γi(t) i − (1/2)

ρfc(t, x) dx =
−∞

N

(1 ≤ i ≤ N).

[They agree with the classical locations of (4.22).]

2376

LEE, SCHNELLI, STETLER AND YAU

PROOF. The random matrix Wt ≡ (wij,t ) := e−t/2W + (1 − e−t )1/2W satisfies the assumptions in Definition 2.1: the entries are centered and have variance
1/N . Moreover, since the distributions of (wij,0), satisfies (2.3) and since (wij ) are real, respectively, complex, centered Gaussian random variables with variance
1/N , respectively, 2/N , the distributions of (wij,t ) also satisfy (2.3). The claim now follows from (3.15) of Corollary 3.4 and the moment bounds E Tr Wt2p ≤ Cp (see, e.g., [1]), as well as the boundedness of (vi).

5.2. Entropy decay estimates. Let ω and ν be two (probability) measures on

RN that are absolutely continuous with respect to Lebesgue measure. We denote

the

Radon–Nikodym

derivative

of

ν

with

respect

to

ω

by

dν dω

,

define

the

relative

entropy of ν with respect to ω by

dν dν

(5.8)

S(ν|ω) :=

log dω,

RN dω dω

and, in case ν = f ω, f ∈ L1(RN ), abbreviate

Sω(f ) = S(f ω|ω).

The entropy Sω(f ) controls the total variation norm of f through the inequality

(5.9) |f − 1| dω ≤ 2Sω(f ),

a result we will use repeatedly in the next sections.

Besides the dynamics (ft )t≥0 generated by L introduced in Section 5.1, we also

consider a (a priori undetermined) time dependent density, (ψ˜ t )t≥0, with respect to

μG. We assume that ψ˜ t = 0, almost everywhere with respect to μG and abbreviate

g˜t

:=

ft ψ˜ t

.

Setting

ω˜ t

:=

ψ˜ t μG,

we

can

write

ft (λ)μG(dλ) = g˜t (λ)ω˜ t (dλ).

A natural choice for ψ˜ t μG is the time dependent β-ensemble, ψt μG, introduced in (4.21). Yet, following the arguments of Erdo˝s et al. [29] we make a slightly different choice for ψ˜ t : for τ > 0, we define a measure ψ˜ t μG on (N) by setting

(5.10)

ψ˜ t (λ)μG(dλ)

:=

1 Zψ˜ t

e−Nβ

Ni=1(λi −γi (t))2/(2τ )ψt (λ)μG(dλ),

where Zψ˜t ≡ Zψ˜t (β) is chosen such that ψ˜ t (λ)μG(dλ) = 1. In the following, we mostly choose τ to be N -dependent with 1 τ > 0.
We call the measure ψ˜ t μG the instantaneous relaxation measure. The density ψ˜ t depends on V = diag(vi) via the initial condition ψ˜ 0. As for the distribution ft , we condition on V and omit this from the notation. We may write the measure ψ˜ t μG in the Gibbs form

ψ˜ t (λ)μG(dλ)

=

1 Zψ˜ t

e−βNHt (λ)

dλ

λ ∈ (N) ,

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2377

with

(5.11)

N
Ht (λ) = HG(λ) +
i=1

(λi − γi(t))2 + U (t, λi) 2τ 2

,

where HG is defined in (4.10) and Zψ˜t ≡ Zψ˜t (β) is a normalization. Then we compute

∇u · ∇2Ht

N
· ∇u ≥ (∂iu)2
i=1

1 + U (t, λi) + 1 τ 22

(5.12)

+

1 N

N i=1

(i) j

(xi

1 − xj

)2

(∂i

u

−

∂j

u)2

≥ N (∂iu)2 , i=1 2τ

for u ∈ C1(RN ) and τ sufficiently small (independent of N ), where we use that U (t, ·) is uniformly bounded below by Lemma 4.2. Then, by the Bakry–Émery criterion [3], there is a constant C such that the following logarithmic Sobolev inequality holds for all sufficiently small τ > 0:

(5.13)

Sω˜ t (q) ≤ Cτ Dω˜ t (√q)

(t ≥ 0),

where q ∈ L∞(dω˜ t ) is such that q dω˜ t = 1. We refer, for example, to [28–30, 33] for more details.
Recall the definition of ψt μG in (4.21). Let Lt denote the generator defined by the natural Dirichlet form with respect to ωt , that is,

(5.14)

1N Dωt (q) = βN i=1

(∂iq)2 dωt = −

qLt q dωt

The main result of this section is the following proposition.

(t > 0).

PROPOSITION 5.3. Let gt := ft /ψt , and set ωt := ψt μG such that

S(ft μG|ψt μG) = Sωt (gt ).

Then there is a constant C (independent of t) such that, for all 0 < a < 1/2, we have

(5.15)

∂t Sωt (gt ) ≤ −4Dωt ( gt ) + CN 1−2a (t > 0),

for N sufficiently large on .

2378

LEE, SCHNELLI, STETLER AND YAU

The results of Proposition 5.3 resemble the relative entropy estimate of Theorem 2.5 in [30] for Wigner matrices. However, due to the fact that both distributions ft μG and ψt μG are not close to the global equilibrium for the Dyson Brownian motion, μG, the reference ensemble ψt μG changes with time, too. Thus to establish (5.15), we need to include additional factors coming from time derivatives of ψt μG. These can be controlled using the definition of the potential U (t). The idea of choosing slowly varying time dependent approximation states and controlling the entropy flow goes back to the work [61].
The relative entropy Sωt and the Dirichlet form Dωt do not satisfy the logarithmic Sobolev inequality (5.13). However, we have for t > 0 the estimates

(5.16) and (5.17) respectively,

Dωt (

gt

)

≤

2Dω˜ t

√ ( g˜t

)

+

C

β

N 2Qt τ2

Dω˜ t (

g˜t ) ≤ 2Dωt (

gt

)

+

C

βN 2Qt τ2

,

(5.18)

Sω˜ t (g˜t ) = Sωt (gt ) + O

βN 2Qt τ

,

where we have set

(5.19)

Qt

:= Eft μG 1 N

N i=1

λi

− γi(t)

2.

Estimates (5.16), (5.17) and (5.18) can be checked by elementary computations, which we omit here. In the following we always bound Qt ≤ CN −1−2a [t ≥ 0, a ∈ (0, 1/2)]; see Lemma 5.6. Using (5.16), (5.17) and (5.18) in combination with the
logarithmic Sobolev inequality (5.13) and wit√h Proposition 5.3, we can follow [30] to obtain a bound on the Dirichlet form Dωt ( gt ).

COROLLARY 5.4. Under the assumptions of Proposition 5.3, the following
holds on for N sufficiently large. For any ε > 0 and t ≥ τ N ε with 1 τ ≥ N −2a, we have the entropy and Dirichlet form bounds

(5.20)

N 1−2a Sωt (gt ) ≤ C τ ,

N 1−2a Dωt ( gt ) ≤ C τ 2 ,

where the constants depend on ε .

Before we prove Proposition 5.3, we obtain rigidity estimates for the time de-
pendent β-ensemble ψt μG. Recall that we denote by (γi(t)) the classical locations with respect to the measure ρfc(t). Also recall the notation αˇ i = min{i, N − i + 1}.

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2379

LEMMA 5.5. Let U (t, ·), t ≥ 0 be as in Lemma 4.2. Then the following holds on for N sufficiently large:
For any δ > 0, there is ς > 0 such that

(5.21)

Pψt μG λi − γi (t ) > N −(2/3)+δαˇ i−1/3 ≤ e−Nς ,

for all t ≥ 0, 1 ≤ i ≤ N , where Pψt μG , stands for the probability under ψt μG conditioned on . Moreover, for any 0 < a < 1/2, we have

(5.22)

sup
t ≥0

1 N

N i=1

λi

− γi(t)

2ψt (λ)μG(dλ) ≤

N −1−2a,

for N sufficiently large.

PROOF. The rigidity estimate (5.21) follows from Proposition 4.3 by choosing N ∈ N sufficiently large. Estimate (5.22) is a direct consequence of (5.21) and the fast decay of the distribution ψt (λ)μG(λ).

For brevity, we often drop the t-dependence of γi(t) from the notation.

PROOF OF PROPOSITION 5.3. Recall that we have set gt = ft /ψt and ωt = ψt μG. The relative entropy S(ft μG|ψt μG) = Sωt (gt ) satisfies [61],

(5.23)

∂t

S(ft

μG|ψt

μG)

=

−

1 βN

|∇ gt gt

|2

ψt

dμG

+

(L

− ∂t ψt

)ψt

ft

dμG.

We note that the first term on the right-hand side of (5.23) equals

(5.24)

−1 βN

|∇ gt gt

|2

ψt

dμG

=

−4Dωt

(

gt ).

To bound the second term on the right-hand side of (5.23), we write

(5.25)

(L

− ∂t ψt

)ψt

ft

dμG

=

(Lt gt )

dωt

+

1 2

N
U (t, λi) ∂igt (λ) dωt (λ) −
i=1

gt ∂t ψt dμG,

with Lt defined in (5.14). Note that the first term on the right-hand side of (5.25) vanishes since, by con-
struction, ωt is the reversible measure for the instantaneous flow generated by Lt . The last term on the right-hand side of (5.25) can be computed explicitly as (recall
that the normalization Zψt in the definition of ψt μG also depends on t),

(5.26)

−

gt ∂t ψt dμG = Eft μG − Eψt μG

βN N

2

∂t U (t, λi) .
i=1

2380

LEE, SCHNELLI, STETLER AND YAU

To deal with the second term on the right-hand side of (5.25), we integrate by parts

to find

1N

2

U (t, λi) ∂igt (λ) dωt (λ)
i=1

(5.27)

= Eft μG

−1 N U 2 i=1

(t, λi)

+ Eft μG

βN N

4

U (t, λi)
i=1

U

(t, λi)

+

λi

−

2 N

(i) j

1 λi − λj

.

Setting gt ≡ 1 in the above computation, we also obtain the identity

0 = Eψt μG

1N −U
2 i=1

(t, λi)

(5.28)

+ Eψt μG

βN N

4

U (t, λi)
i=1

U

(t, λi)

+

λi

−

2 N

(i) j

1 λi − λj

.

Equation (5.28) may alternatively be derived from the “first order loop equation”

for the β-ensemble ψt μG. Equation (5.27) can thus be rewritten as

1N

2

U (t, λi) ∂igt (λ) dωt (λ)
i=1

(5.29)

= Eft μG − Eψt μG

−1 N U 2 i=1

(t, λi)

+ Eft μG − Eψt μG

×

βN N

4

U (t, λi)
i=1

U

(t, λi)

+

λi

−

2 N

(i) 1 j λi − λj

.

Next, to control the second and third terms on the right-hand side of (5.25), respectively, the right-hand side of (5.29), we proceed as follows. We expand the

potential terms U (t, λi), respectively, U (t, λi), in Taylor series in λi to second

order around the classical location γi. The resulting zero order terms cancel ex-

actly since the classical locations of the ensembles ft μG, and ψt μG agree by con-

struction. The first order terms in the Taylor expansion can (1) either be bounded

in terms of the expectations of

N i=1

(λi

−

γi )2

(which

can

be

controlled

with

the

rigidity estimates in Lemmas 5.5 and 5.2); or (2) they cancel exactly due to the def-

inition of the potential U (t, ·) and its equation of motion in (4.27). Finally, the sec-

ond order terms in the Taylor expansion can be bounded by the rigidity estimates

in Lemmas 5.5 and 5.2. The details are as follows.

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2381

Expanding ∂t U (t, λi) to second order around γi , we obtain from (5.26) that − gt ∂t ψt dμG = Eft μG − Eψt μG

(5.30)

×

βN 2

N
∂t U (t, γi) +
i=1

βN 2

N
∂t U
i=1

(t, γi)(λi

−

γi )

+ O N 1−2a ,

on , where we use the rigidity estimates in Lemmas 5.5 and 5.2, and that
∂t U (t, ·) is uniformly bounded on compact sets by Lemma 4.2. To save notation, we introduce a function G : R+ × R2 → R by setting

(5.31)

G(t; x, y) :=

U

(t, x) − U x−y

(t, y) ,

with G(t; x, x) := U (t, x). Note that G(t; x, y) = G(t; y, x) and that G is C2 in the spatial coordinates by Lemma 4.2. Recalling the equation of motion for ∂t U (t, ·) in (4.27), we can write

(5.32)

∂t U (t,

x)

=

−

U

(t ,

y) dρfc(t, y−x

y)

=

U

(t, y) y

− −

U x

(t, x)

dρfc(t, y)

+

U

(t, x)−

dρfc(t, y) , y−x

for x inside the support of the measure ρfc. Thus, recalling (4.17) and (5.31), we obtain

(5.33)

∂t U (t, x) =

G(t; x, y) dρfc(y) −

1 U
2

(t, x)

U

(t, x) + x

,

for x inside the support of the measure ρfc. We hence obtain from (5.30) that

− gt ∂t ψt dμG

(5.34)

= Eft μG − Eψt μG

βN N 2 i=1

G (t; γi, y) dρfc(y)(λi − γi)

− Eft μG − Eψt μG

βN N

4

U
i=1

(t, γi) U (t, γi) + γi (λi − γi)

− Eft μG − Eψt μG

βN N

4

U (t, γi) U
i=1

(t, γi) + 1 (λi − γi)

+ O N 1−2a ,

2382

LEE, SCHNELLI, STETLER AND YAU

on , where we denote by G (t; x, y) the first derivative of G(t; x, y) with respect

to x.

Next we return to (5.29). Using the rigidity estimates of the Lemmas 5.5 and 5.2,

we find

1N

2

U (t, λi) ∂igt (λ) dωt (λ)
i=1

= Eft μG − Eψt μG

−1 N U 2 i=1

t, γi(t)

(5.35)

+ Eft μG − Eψt μG

×

βN N

4

U (t, λi)
i=1

U

(t ,

λi )

+

λi

−

2 N

(i) j

1 λi − λj

+ O N 1/2−a ,

on , where we use a Taylor expansion of the first term on the right-hand side

of (5.29). Here we also use that U is three times continuously differentiable with uniformly bounded derivatives on compact sets. Note that the first term on the right-hand side of (5.35) vanishes.
Using the definition of G(t; ·, ·) in (5.31), we can recast (5.35) as

1N

2

U (t, λi) ∂igt (λ) dωt (λ)
i=1

(5.36)

= Eft μG − Eψt μG

βN 4

N
U
i=1

(t, λi)

U

(t, λi) + λi

+ Eft μG − Eψt μG

βN −

N

1

4 i=1 N

(i)
G(t; λi, λj )
j

+ O N 1/2−a ,

where we use the symmetry G(t; x, y) = G(t; y, x). Expanding the second term on the right-hand side (5.36) to second order in (λi, λj ) around (γi, γj ), we obtain

Eft μG − Eψt μG

−βN N 1 4 i=1 N

(i)
G(t; λi, λj )
j

(5.37)

= Eft μG − Eψt μG

−βN N 2 i=1

1 N

N
G
j =1

(t; γi, γi)

(λi − γi)

+ O N 1/2−a + O N 1−2a ,

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2383

on , where we use G(t; x, y) = G(t; y, x), G(t; x, x) = U (t, x) and that G(t; x, y) is C2 in the spatial variables. Thus, also expanding the first term on
the right-hand side of (5.36) in λi around γi , we obtain

1 2
(5.38)

N
U (t, λi) ∂igt (λ) dωt (λ)
i=1

= Eft μG − Eψt μG

βN N 1 − 2 i=1 N j G (t; γi, γj ) (λi − γi)

+ Eft μG − Eψt μG

βN N

4

U
i=1

(t, γi) U (t, γi) + γi (λi − γi)

+ Eft μG − Eψt μG

βN N

4

U (t, γi) U
i=1

(t, γi) + 1 (λi − γi)

+ O N 1/2−a + O N 1−2a ,

on , where we use the rigidity estimates in Lemmas 5.5 and 5.2. Adding up (5.34) and (5.38), we hence obtain

(L

− ∂t ψt

)ψt

ft

dμG

≤ βN Eft μG 2

N
−
i=1

1N

N

G
j =1

(t; γi, γj )

− G (t; γi, y) dρfc(y) (λi − γi)

− Eψt μG

N
−
i=1

1 N

N
G
j =1

(t; γi, γj )

− G (t; γi, y) dρfc(y) (λi − γi)

+ O N 1/2−a + O N 1−2a ,

on . To finish the proof we observe that for all γi,

1 N

N
G
j =1

(t; γi, γj )

=

G (t; γi, y) dρfc(t, y) + O N −1 ,

2384

LEE, SCHNELLI, STETLER AND YAU

on , where we use that γi+1 − γi ∼ N −2/3αˇ i−1 (αˇ i = min{i, N − i + 1}), and the square root decay of ρfc(t) at the edges of the support. Thus

(5.39)

(L − ∂t )ψt ψt

ft

dμG

=

O

N 1/2−a

+ O N 1−a ,

for N sufficiently large on , where we use one last time the rigidity estimates. Using that N 1/2−a < N 1−2a, a ∈ (0, 1/2), we get from (5.23), (5.24) and (5.39) the desired estimate (5.15).

Before we move on to the proof of Corollary 5.4, we give a rough estimate on Sωt (gt ) for t > 0.

LEMMA 5.6. There is a constant m such that, for τ > 0 and t ≥ τ , we have

(5.40)

Sωt (gt ) = S(ft μG|ψt μG) ≤ CN m

on , for N sufficiently large. Here the constant C depends on τ .

PROOF. From the definition of the relative entropy in (5.8), we have

S ft μG|ψt μG

(5.41)

≤ S ft μG|μG

+

βN N 2 i=1

U (t, λi)ft (λ) dμG(λ) + log Zψt .

Since the potential U (t) is bounded below, we have (for N sufficiently large on ) log Zψt ≤ CβN 2. Similarly, using the rigidity estimate (5.6), we can bound the
second term on the right-hand side of (5.41) by CN2. To bound the first term on the right of (5.41), we use that S(ft μG|μG) ≤ S(Ht |W ) ≤ N 2 max S(hij,t |wij ) + N max S(hii,t |wii), where (hij,t ) are the entries of the in (5.2) and wij are the en-
tries of the GOE, respectively, GUE, matrix W . By explicit calculations, remem-
bering that the diagonal entries (vi) are fixed, one finds max S(hij,t |gij ) ≤ CN for t ≥ τ ; see, for example, [22]. (Note that we choose t > 0; otherwise the relative
entropy may be ill defined.)

To complete the proof of Corollary 5.4 we follow the discussion in [30].

PROOF OF COROLLARY 5.4. Using an approximation argument, we can assume that g˜t ∈ L∞(dω˜ t ). Using first the entropy bound (5.15) and then the Dirichlet form estimate in (5.17), we obtain

∂t Sωt (gt ) ≤ −4Dωt ( gt ) + CN 1−2a

≤

−2Dω˜ t

√ ( g˜t

)

+

CN

1−2a

+

C

N 1−2a τ2

≤

−C

τ

−1Sω˜ t

(g˜t

)

+

C

N

1−2a
τ2

,

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2385

for N sufficiently large on . To get the third line we use the logarithmic Sobolev inequality (5.13) and that, by assumption, τ < 1. Using the entropy estimate (5.18), we thus obtain

(5.42)

∂t

Sωt

(gt

)

≤

−C

τ

−1

Sωt

(gt

)

+

C

N

1−2a
τ2

,

for N sufficiently large on . Integrating (5.42) from τ to t/2, we infer

Sωt

/2

(gt

/2)

≤

e−Cτ

−1

(t

/2−τ

)

Sωτ

(gτ

)

+

C

N

1−2a
τ

,

for N sufficiently large on . Bounding Sωτ (gτ ) by (5.41), we get

Sωt /2

(gt /2 )

≤

CN

me−Cτ

−1 (t /2−τ

)

+

C

N

1−2a
τ

,

for N sufficiently large on . Recalling that t ≥ τ0 = τ N ε and using the monotonicity of the relative entropy, we obtain the first inequality in (5.20).
Integrating (5.15) from t/2 to t, we obtain

tt
t/2 Dωs ( gs ) ds ≤ − t/2 ∂s Sωs (gs ) ds + Ct N 1−2a.

Thus, using the above estimate on the relative entropy and the monotonicity of the Dirichlet form,

Dωt (

gt )

≤

N 1−2a C
tτ

+

CN 1−2a.

Recalling that t ≥ τ0 = N ε τ , we get the second inequality in (5.20).

6. Local equilibrium measures. The estimates on the relative entropy and the Dirichlet form obtained in Corollary 5.4 do not directly imply that the local statistics of the measures ft μG and ψt μG agree in the limit of large N . However, the averaged local gap statistics of ft μG, ψt μG and μG can be compared (for 1 t N −1/2) for large N as is asserted in the main theorems of this section, Theorem 6.1 and Theorem 6.2, below. We first state these results and give a short outline of their proofs in Section 6.1 before going into the details in Sections 6.2–6.5.

6.1. Averaged local gap statistics for small times. Recall that we call a symmetric function O : Rn → R, n ∈ N, an n-particle observable if O is smooth and
compactly supported. For a given observable O, a time t ≥ 0, a small constant
α > 0 and j ∈ [[αN, (1 − α)N ]], we define an observable Gj,n,t (x) ≡ Gj,n(x), by setting

(6.1)

Gj,n(x) := O Nρj (xj+1 − xj ), Nρj (xj+2 − xj ), . . . , Nρj (xj+n+1 − xj ) ,

2386

LEE, SCHNELLI, STETLER AND YAU

x = (xk)Nk=1 ∈ (N), where we set Gj,n = 0 if j + n > (1 − α)N . Here ρj denotes the density of the measure ρfc(t) at the classical location of the j th particle at time t , that is, ρj := ρfc(t, γj (t)). We also set

(6.2)

Gj,n,sc(x) := O Nρsc,j (xj+1 − xj ), Nρsc,j (xj+2 − xj ), . . . , Nρsc,j (xj+n+1 − xj ) ,

x ∈ (N), where ρsc,j denotes the density of the semicircle law at the classical location of the j th particle with respect to the semicircle law.
In the following, we denote constants depending on O by CO . Recall the definition of the density ψt in (4.21). We have the following statement on the averaged local gap statistics.

THEOREM 6.1. Let n ∈ N be fixed, and consider an n-particle observable O.
Fix a small constant α > 0, and consider an interval of consecutive integers J ⊂
[[αN, (1 − α)N]] in the bulk. Then, for any small δ > 0, there is a constant f > 0 such that, for t ≥ N −1/2+δ,

(6.3)

1 |J | j∈J Gj,n(x)ft (x) dμG(x) −
≤ CO N −f,

1 |J | j∈J Gj,n(x)ψt (x) dμG(x)

for N sufficiently large on . The constant CO depends on α and O, and the constant f depends on α and δ.

We can also compare the averaged local gap statistics of ft μG, with the averaged local gap statistics of the Gaussian unitary, respectively, orthogonal, ensem-
ble.

THEOREM 6.2. Under the same assumptions as in Theorem 6.1 and with similar constants, we have

1 |J

|

j

∈J

Gj,n(x)ft

(x)

dμG

(x)

−

1 |J | j∈J Gj,n,sc(x) dμG(x)

≤ CO N −f,

for N sufficiently large on .

The proofs of Theorem 6.1 and Theorem 6.2 proceed in two steps. We first localize the measures ft μG and ψt , μG; that is, we study the statistics of K, 1 K N , consecutive particles inside the bulk—the interior particles—with the remaining particles—the exterior particles—being fixed; for details, see Section 6.2. For most configurations of the exterior particles (boundary conditions), we can compare the statistics of the localized versions of ft μG and ψt , μG. This

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2387

is accomplished in Proposition 6.4 of Section 6.3 by using that (1) the localized β-ensemble satisfies a logarithmic Sobolev inequality (6.23) with constant CK/N and that (2) the localized Dirichlet form can be controlled by the global Dirichlet form [see (6.24)], the latter being estimated in Corollary 5.4.
In a second step, we use Theorem 4.1 of [31] that, roughly speaking, assures that the local gap statistics of localized β-ensembles are essentially independent of the boundary conditions and indeed agree with the local gap statistics of the Gaussian ensembles. Putting this universality result to work in Section 6.4, we conclude that the local gap statistics of the localized version of the measure ft μG are universal, for 1 t N −1/2 and for most boundary conditions. Theorems 6.1 and 6.2 are then proven in Section 6.5 by integrating out the boundary conditions.
We conclude this subsection with the following two remarks: once the entropy estimate of Proposition 5.3 has been established, one can apply the methods of [31] to prove the gap universality in the bulk for deformed Wigner matrices; see Remark 2.8 above for an explicit statement; we leave the details to the interested readers.
As an alternative to the approach outlined above, one could combine the approach from [30] with Theorem 2.1 in [12] (see Theorem 4.1 above), to prove Theorems 6.1 and 6.2.

6.2. Preliminaries. Let α, σ > 0 be two small positive numbers, and choose two integer parameters L and K such that

(6.4)

L ∈ αN, (1 − α)N , K ∈ N σ , N 1/4 .

We denote by IL,K := [[L − K, L + K]] a set of K := 2K + 1 consecutive indices in the bulk of the spectrum. Below we often abbreviate I ≡ IL,K . Recall the definition of the set (N) ⊂ RN in (4.1). For λ ∈ (N), we write

(6.5)

λ = (y1, . . . , yL−K−1, xL−K , . . . , xL+K , yL+K+1, . . . , yN ),

and we call λ a configuration (of N particles or points on the real line). Note that on the right-hand side of (6.5) the points keep their original indices and are in increasing order so that

(6.6)

x = (xL−K , . . . , xK+L) ∈ (K), y = (y1, . . . , yL−K−1, yL+K+1, . . . , yN ) ∈ (N−K).

We refer to x as the interior points or particles and to y as the exterior points or
particles.
In the following, we often fix the exterior points and consider the conditional measures on the interior points: let ω be a measure on (N) with a density. Then we denote by ωy the measure obtained by conditioning on y; that is, for λ in the
form of (6.5),

ωy(dx) ≡ ωy(x) dx :=

ω(λ) dx

=

ω(x, y) dx ,

ω(λ) dx ω(x, y) dx

2388

LEE, SCHNELLI, STETLER AND YAU

where, with slight abuse of notation, ω(x, y) stands for ω(λ). We refer to the fixed exterior points y as boundary conditions of the measure ωy. For fixed y ∈ (N−K), all (xi) lie in the open configuration interval

I ≡ IL,K := (yL−K−1, yL+K+1).

Set y¯ := (yL−K−1 + yL+K+1)/2, and let

(6.7)

αj

:=

y¯

+

j −L K+1

|y|

(j ∈ IL,K )

denote K equidistant points in the interval I. Let U ∈ C4(R) be a “regular” potential satisfying (4.4) and (4.5). We then con-

sider the β-ensemble

(6.8)

μ(dλ)

≡

μU

(dλ)

:=

1 ZU

e−β N H(λ)

dλ

with [cf. (4.2)]

(β > 0),

(6.9)

H(λ) := N 1 i=1 2

U (λi)

+

λ2i 2

−

1 N

1≤i<j ≤N

log |λj

−

λi |,

and with ZU ≡ ZU (β) a normalization. For K, L and y fixed, we can write μy as the Gibbs measure

(6.10)

μy(dx)

=

1 ZUy

e−β N Hy (x)

dx,

where

(6.11)

Hy(x)

=

i∈I

1 2

V

y(xi

)

−

1 N

i,j ∈I

log |xj

− xi|,

i<j

with

(6.12)

V y(x) = U (x) +

x2 2

−

2 N

i∈/I

log |x − yi|

an external potential and with ZUy ≡ ZUy (β) a normalization. Following [31], we next introduce the notion of regular external potential:

DEFINITION 6.3. An external potential V ≡ V y of a β-ensemble of K points in a configuration interval I = (a, b) is called Kχ -regular if the following bounds

hold: (6.13)

|I| =

K

Kχ +O ,

Nρ (y¯ )

N

(6.14)

V (x) = ρ(y¯) log d+(x) + O Kχ ,

d−(x)

N d (x )

(6.15)

c V (x) ≥ 1 + inf U (x) + ,
d (x )

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2389

for x ∈ I, with some c > 0 and for some small χ > 0, where d(x) := min |x − a|, |x − b|
denotes the distance to the boundary of I, d−(x) := d(x) + ρ(y¯)N −1Kχ
and d+(x) := max |x − a|, |x − b| + ρ(y¯)N −1Kχ .

The main technical result we use in this section is Theorem 4.1 of [31]; see Theorem 6.5 below. It asserts that the local gap statistics of μy are essentially independent of y and U , provided that V y is Kχ -regular for some small χ > 0.

6.3. Comparison of local measures. Fix small α, σ > 0, and let K and L sat-
isfy (6.4). Recall that we denote by ft μG the distribution of the eigenvalues of the matrix in (5.2) and by ψt μG the reference β-ensemble defined in (5.10). Following the discussion in Section 6.2, we introduce the conditioned densities

(6.16)

ftyμyG = (ft μG)y,

ψtyμyG = (ψt μG)y.

Recall that we denote by ρfc(t) the equilibrium density of ψt μG and by γk ≡ γk(t) the classical location of the kth particle with respect to ρfc ≡ ρfc(t); cf. (3.14). Let ε0 > 0 and define the set of “good” boundary conditions, RL,K ≡ RL,K (ε0, α),

RL,K : = λ ∈ (N) : |λk − γk| ≤ N −1+ε0 , ∀k ∈ αN, (1 − α)N \ IL,K (6.17)
∩ λ ∈ (N) : |λk − γk| ≤ N −2/3+ε0 , ∀k ∈ [[1, N ]] .
The next result compares the local statistics of ftyμyG and ψtyμyG for y ∈ RL,K . Recall that a stands for any number in (0, 1/2).

PROPOSITION 6.4. Fix small constants α, σ > 0 [see (6.4)] and ε0 > 0;
see (6.17). Let K satisfy (6.4), and let O be an n-particle observable. Let ε > 0, and choose τ satisfying 1 τ > N −2a. Then, for any t ≥ N ε τ and any constant
c ∈ (0, 1), there is a set of configurations G ≡ GL,K (ε0, α) ⊂ RL,K (ε0, α), with

(6.18)

Pft μG (G) ≥ 1 − N −c , 2

such that (6.19)

O (x)

fty(x) − ψty(x)

μyG(dx)

≤

CO

√ K

N

c−aτ

−1,

t ≥ N ε τ , for N sufficiently large on . The constant CO , depends only on ε , α and O.

2390

LEE, SCHNELLI, STETLER AND YAU

Moreover, there is υ > 0, such that (6.20) PftyμyG xk − γk(t ) < N −1+ε0 , k ∈ IL,K ≥ 1 − e−υ(ϕN )ξ , t ≥ N ε τ , for N sufficiently large on , with ξ = A0 log log N/2; see (2.20).

PROOF. We follow closely the proof of Lemma 6.4 in [31]. Let τ satisfy 1 τ > N −2a, and choose t ≥ N ε τ . We estimate

(6.21)

O(x) fty(x) − ψty(x) μyG(dx) ≤ CO ftyμyG − ψtyμyG 1 ≤ CO SψtyμyG gty ,

where we use (5.9) and set gt := ft /ψt . For y ∈ RL,K , we consider the locally constrained measure ψtyμyG, explicitly given by

ψtyμyG(dx)

=

1 Zy

e−Nβ

Hy

(t

,x)

dx,

with

Hy(t, x) =

U (t, xk) + xk2

k∈I 2

4

11 − N k,l∈I log |xk − xl| − N k∈I log |xk − yl|.
k<l l∈/I

Here I ≡ IL,K . From (5.20) of [31], we know that

(6.22)

∇x2Hy(t, x) ≥ cN/K (y ∈ RL,K ),

for some c > 0 independent of t. Here, ∇x2 denotes the Hessian with respect the variables x. Thus the Bakry–Émery criterion yields the logarithmic Sobolev in-
equality

(6.23)

SψtyμyG

gty

CK ≤ N DψtyμyG

gty

(y ∈ RL,K ),

where the constant C can be chosen independent t.

For k ∈ [[1, N ]], denote by Dψt μG,k the Dirichlet form of the particle k, that is,

Dψt μG,k(f

)

:=

1 2N

|∂kf |2ψt μG, and by DψtyμyG,k its conditioned analogue (with

k ∈ IL,K ). Using the notation of (6.5), we may write

Eft μG DψtyμyG gty = DψtyμyG gty ft (λ)μG(dλ),

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2391

and we can bound Eft μG DψtyμyG
(6.24)

gty = Eft μG

DψtyμyG,k

k∈I

≤ Dψt μG ( gt ) ≤ CN 1−2aτ −2,

gty

for N sufficiently large, where we use Corollary 5.4 in the last line. Thus Markov’s inequality implies, for c > 0, that there exists a set of configurations G1 ⊂ R, with Pft μG(G1) ≥ 1 − N −c, such that, for y ∈ G1,

(6.25)

DψtyμyG gty ≤ CN 2cN 1−2aτ −2

holds for N sufficiently large on . Substituting (6.25) into (6.23) and then

into (6.21), we find that

O (x)

fty − ψty

μyG(dx)

≤

CO

√ K

N

cN

−aτ

−1,

on for N sufficiently large. This proves (6.19). To prove (6.20) note that the rigidity estimates of Lemma 5.6 imply
Eft μG PftyμyG xk − γk(t ) > N −1+ε, k ∈ I
= Pft μG xk − γk(t ) > N −1+ε, k ∈ I ≤ e−υ(ϕN )ξ ,
for some υ > 0, where we have chosen ξ = A0 log log N/2. By Markov’s inequality we conclude that there is a set of configurations, G2, such that (6.20) holds with (ξ, υ)-high probability. Finally, set G := G1 ∩ G2, and note that G satisfies (6.18).

6.4. Gap universality for local measures. In Section 6.3, we show that the local gap statistics of the measure ftyμyG agree with those of ψtyμyG for boundary conditions y in the set RL,K . In this subsection, we are going to show that the local statistics of ψtyμyG are essentially independent of the precise form of y, as is asserted by the main theorem of [31]. Recall the notion of external potential
introduced in (6.12).

THEOREM 6.5 (Gap universality for local measures, Theorem 4.1 in [31]). Let L, L and K = 2K + 1 satisfy (6.4) with α, σ > 0. Consider two boundary conditions y, y˜ such that the configuration intervals coincide, that is,

(6.26)

I = (yL−K−1, yL+K+1) = (y˜L−K−1, y˜L+K+1).

Consider two measures μ and μ˜ in the form of (6.8), with possibly two different potentials U and U , and consider the constrained measures μy and μ˜ y˜ . Let χ > 0,

2392

LEE, SCHNELLI, STETLER AND YAU

and assume that the external potentials V y and V˜ y˜ [see (6.12)] are Kχ -regular;

see Definition 6.3. In particular, assume that I satisfies

(6.27)

K Kχ

|I| = N

U (y¯) + O

N

.

Assume further that

(6.28)

max
j ∈IL,K

Eμy xj

− αj

+ max
j ∈IL˜ ,K

Eμ˜ y˜ xj

− αj

≤ CN −1Kχ .

Let p ∈ Z satisfy |p| ≤ K − K1−χ , for some small χ > 0. Fix n ∈ N. Then there is a constant χ0, such that if χ , χ < χ0, then for any n-particle observable O, we have
Eμy O N (xL+p+1 − xL+p), . . . , N (xL+p+n − xL+p)

− Eμ˜ y˜ O N (xL+p+1 − xL+p), . . . , N (xL+p+n − xL+p) ≤ CO K−b,

for some constant b > 0 depending on σ , α, and for some constant CO depending on O. This holds for N sufficiently large [depending on the χ , χ , α and C
in (6.28)].

Recall that the measure ψtyμyG can be written as the Gibbs measure

(6.29)

ψtyμyG(dx)

=

1 Zψy t

e−Nβ Hy (t ,x)

dx,

where

(6.30)

Hy(t, x)

=

i∈I

1 2

V

y(t

,

xi

)

−

1 N

i,j ∈I

log |xj

− xi|,

i<j

with the external potential

(6.31)

V

y(t, x)

=

U (t, x)

+

x2 2

−

2 N

i∈/I

log |x

−

yi |.

Using Theorem 6.5 we first show that the local statistics of ψtyμyG are virtually independent of y; that is, we apply Theorem 6.5 with μy = (ψt μG)y and μ˜ y˜ = (ψt μG)y˜ .
We first check the regularity assumption of the external potential V y. Recall the

definition of Kχ -regular potential in Definition 6.3.

LEMMA 6.6. Fix small constants α, σ > 0; see (6.4). Let χ > 0, and consider y ∈ RL,K (χ σ/2, α/2). Then, on the event , the external potential V y(t, x) in (6.31) is Kχ -regular on I = (yL−K−1, yL+K+1).

The proof of Lemma 6.6 follows almost verbatim the proof of Lemma 4.5 in the Appendix A of [31], and we therefore omit it here.

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2393

To check that assumption (6.28) of Theorem 6.5 holds, we use the following result. Recall the set of configurations G of Proposition 6.4.

LEMMA 6.7. Under the assumptions of Proposition 6.4 the following holds. Let y ∈ G. Then, for all k ∈ IL,K ,

(6.32)

EftyμyG xk − EψtyμyG xk

≤ C KN 2c N −aτ −1 N

t ≥ τNε ,

for N sufficiently large on .

PROOF. We follow the proof of Lemma 6.5 of [31]. Fix t ≥ τ N ε , where 1 τ ≥ N −2a. Let y ∈ G. Denote by Lyt the generator associated to the Dirichlet form DψtyμyG , that is,

f Lyt gψty dμyG

=− 1 βN

i∈I

∂i f ∂i gψty dμyG

(I ≡ IL,K ).

Let qs be condition

the solution of the q0 := gty = fty/ψty.

evolution equation ∂sqs = Lyt qs , s ≥ 0, with initial Note that qs is a density with respect to the reversible

measure, ψtyμyG, of this dynamics. Hence, we can write

EftyμyG xk − EψtyμyG xk =

∞
ds

0

xkLyt qs ψty dμyG

=1

∞
ds

βN 0

∂kqs ψty dμyG .

Recall that ψtyμyG satisfies the logarithmic Sobolev inequality (6.23) with con-

stant τK and the

:= CK/N , provided that y ∈ RL,K . exponential decay of the Dirichlet

Thus, form

DupψotynμyGu(s√ingqsC),auwcehyo–bStcahinwaforzr

some υ , c > 0,

EftyμyG xk − EψtyμyG xk

=

1 βN

N υ τK
ds
0

∂kqs ψty dμyG + O e−cNυ .

Using

|∂k qs

|

=

2|√qs

∂k

√ qs

|

≤

√ R(∂k qs

)2

+

R−1qs

,

where R > 0 is a free parameter, we obtain

1 Nυ dsτK βN 0

∂kqs ψty dμyG

≤R

N υ τK

√

0 ds DftyμG ( qs )

+

1 R−1N −1+υ 2

τK

2394

LEE, SCHNELLI, STETLER AND YAU

≤ RSψtyμyG

gty

+

1 R−1N −1+υ 2

τK

≤ CRτK DψtyμyG

gty

+

1 R−1N −1+υ 2

τK ,

where in the second line we use that the time integral of the Dirichlet form is bounded by the initial entropy (see, e.g., Theorem 2.3 in [30]) and in the final line we used the logarithmic Sobolev inequality (6.23). Optimizing over R, we get

EftyμyG xk − EψtyμyG xk ≤ CτK N −1+υ DψtyμyG gty 1/2 + O e−cNυ

KN υ /2 ≤ C N 3/2 DψtyμyG

gty 1/2 + O e−cNυ ,

where we used that τK = CK/N . Using (6.25) we finally obtain

EftyμyG xk − EψtyμyG xk

≤ C KN 2c N −aτ −1 + O e−cNυ N

,

for N sufficiently large on .

LEMMA 6.8. Fix small constants α, σ > 0. Fix ε > 0 and t ≥ τ N ε , where τ satisfies 1 τ ≥ N −2a. Fix n ∈ N, and consider an n-particle observable O.
Let χ , χ > 0, with χ , χ < χ0, where χ0 is the constant in Theorem 6.5. Then the following holds.
Assume that 0 < a < 1/2, 0 < c < 1, N −2a ≤ τ 1 and K ∈ [[N σ , N 1/4]] are
chosen such that

(6.33)

KN 2c N −aτ −1 ≤ Kχ . NN

Let

p

be

an

integer

satisfying

|p|

≤

K

−

K 1−χ

.

Let

y

∈

GL,K

(

χ

2σ 2

,

α 2

).

Then,

for

the observable G, as defined in (6.1), we have

(6.34)

GL+p,n(x) fty dμyG − ψt dμG

≤

CO

K

−b

+

CO

√ K

N

c

N

−aτ

−1,

for N sufficiently large on , where the constant CO depends on O and ε , and the constant b > 0 depends on α and σ .

PROOF. We follow [31]. Fix t ≥

GL,K

(

χσ 2

,

α).

Then

by

Proposition

6.4

τNε and

and χ > 0. Let the assumption in

y ∈ GL,K (6.33),

(

χ2σ 2

,

α)

⊂

EftyμyG xk − γk(t ) ≤ CKχ N −1,

for all k ∈ I ≡ IL,K . Further, from Lemma 6.7 and the assumption in (6.33) we get

(6.35)

EψtyμyG xk − γk(t ) ≤ CKχ N −1,

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2395

for

all

k

∈

I

.

Recall

from

(6.7)

that

we

denote

by

y¯

:=

1 2

(yL−K

−1

+

yL+K +1 )

the

midpoint of the configuration interval I and that (αk) denote 2K + 1 equidistant

points in I. As shown in Lemmas 4.5 and 5.2 of [31], we have

γk(t) − αk ≤ CKχ N −1,

for

all

k

∈

I

,

provided

that

y

∈

GL,K

(

χσ 2

,

α).

We

hence

obtain

(6.36)

EψtyμyG xk − αk ≤ CKχ N −1,

for N sufficiently large on .

Proposition 6.4 implies that there is CO such that

GL+p,n(x) fty dμyG − ψty dμyG

≤

CO

√ K

N

cN

−a

τ

−1

(6.37)

t ≥ Nε τ ,

for

y

∈

GL,K

(

χσ 2

, α),

N

sufficiently

large

on

.

For α, ε0, ς1 > 0 and a β-ensemble μ on (N), define a set of particle configu-

rations R∗μ ≡ R∗μ(ε0, α) by

R∗μ := y ∈ (N−K) : Pμy |xk − γk| > N −1+ε0 ≤ e−(1/2)Nς1 , ∀k ∈ IL,K ,

where γk denotes the classical location of the kth particle with respect to the equi-

librium measure of μ.

As in the proof of Proposition 6.4, it follows from Markov’s inequality and

the rigidity bound for the β-ensemble ψt μG in Lemma 5.5 that we can choose

R∗ψt μG ⊂ RL,K and that Pψt μG (R∗ψt μG ) ≥ 1 − ce−(1/2)Nς1 , for some c > 0, pos-

sibly

after

decreasing

ς1

by

a

small

amount.

For

y˜

∈

R∗ψt

μG

(

χ

2σ 2

,

α 2

),

Lemma

5.1

of [31] implies that

(6.38)

Eψty˜ μyG˜ xk − αk ≤ CKχ N −1,

for N sufficiently large on . Thus together with (6.36), we have on

(6.39)

Eψty˜ μyG˜ xk − αk + EψtyμyG xk − αk ≤ CK χ N −1,

for

N

sufficiently

large,

for

all

y

∈

G

(

χσ 2

, α)

and

all

y˜

∈

R∗ψt

μG

(

χ

2σ 2

,

α 2

).

We now apply Theorem 6.5: let y˜ and y be as above. By the scaling argument

of Lemma 5.3 in [31], we can assume that the two configuration intervals I˜ and I

agree, so that assumption (6.26) of Theorem 6.5 holds. Moreover, by Lemma 6.6 we know that V y and V y˜ are Kχ -regular external potentials. The assumption

in (6.28) of Theorem 6.5 is satisfied by (6.39). Thus Theorem 6.5 implies that

there is b > 0, depending on σ and α, such that

(6.40)

GL+p,n(x) ψty dμyG − ψty˜ dμyG˜ ≤ CO K−b,

2396

LEE, SCHNELLI, STETLER AND YAU

for N sufficiently large on . Since estimate (6.40) holds for all y˜ ∈ R∗ψtμG , and since Pψt μG (R∗ψt μG) ≥ 1 − e−(1/2)Nς1 , we can integrate over y˜ to find that

GL+p,n(x) ψty dμyG − ψt dμG ≤ CO K−b,

for N sufficiently large on . In combination with (6.37), this yields (6.34).

6.5. Proof of Theorems 6.1 and 6.2. Lemma 6.8 compares the local statistics of the locally-constrained measure ftyμyG with the β-ensemble ψt μG. In order to compare with local statistics of the measure ft μG with ψt μG, we next integrate
out the boundary conditions y.

LEMMA 6.9. Under the assumptions of Lemma 6.8 the following holds. Let J ⊂ [[αN, (1 − α)N ]] be an interval of consecutive integers in the bulk. Then

(6.41)

1 |J |

j ∈J

Gj,n(x)(ft

dμG

−

ψt

dμG)

≤ CO

N −c + K−b + K−χ /2

+

CO

√ K

N

cN

−a

τ

−1,

for N sufficiently large on .

PROOF. For a small χ > 0 as in Lemma 6.8, set K˜ := K − K1−χ /2. We first assume that J is such that |J | ≤ 2K˜ + 1. We then choose L such that J ⊂ IL,K˜ ⊂ ImLe,Kas.uRreecfatyllμtyGhewseeteostfimcoantefigurations G in Proposition 6.4. Using the conditioned

(6.42)

Eft μG

1 |J | j∈J Gj,n

= Eft μG 1 |J |

Gj,n(x)fty dμyG1(G) + O N −c ,
j ∈J

where we used (6.18). Next, using Lemma 6.8 we obtain on

1 |J |

Gj,n(x)fty(x) dμyG(x)1(G)
j ∈J

=1 |J |

Gj,nψt dμG + O K−b

+O

√ K

N

c

N

−aτ

−1

,

j ∈J

on . For the special case |J | ≤ 2K˜ + 1, this yields (6.41). If |J | ≥ K˜ + 1, there are La ∈ [[αN, (1 − α)N ]], with a ∈ [[1, M0]], such that
the intervals ILa,K = [[La − K, La + K]] are nonintersecting with the properties

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2397

that J ⊂

M0 a=1

ILa

,K

and

J

∩ ILa,K

= ∅,

for

all

a

∈ [[1, M0]].

Note

that

M0

≤

|J | K

+ 2.

For

simplicity

of

notation

we

abbreviate

I

(a)

≡

ILa ,K

=

[[La

−K,

La

+ K ]]

and I˜(a) ≡ [[La − K˜ , La + K˜ ]]. We also label the interior and exterior points of a

configuration λ ∈ (N) accordingly,

x(a) = (xLa−K , . . . , xK+La ) ∈ (K),

respectively,

y(a) = (y1, . . . , yLa−K−1, yLa+K+1, . . . , yN ) ∈ (N−K);
cf. (6.6). We let G(a) ≡ GLa,K (ε0, α) ⊂ RLa,K (ε0, α) denote the set of configurations obtained in Proposition 6.4. Using this notation we can write

Eft μG

1 |J | j∈J Gj,n

(6.43)

=1

Eft μG

|J | a : I (a)∩J =∅

+ O N −c ,

Gj,n x(a) fty(a) dμyG(a) 1 G(a)
j ∈I (a)∩J

on , where the first summation on the right-hand side is over indices a ∈ [[1, M0]] such that the intervals (I (a)) satisfy I (a) ∩ J = ∅. Here, we also use the probability estimate on G(a) in (6.18). In (6.43) we may further restrict, for each a, the summation over the index j from I (a) to I˜(a) at an expense of an error term of order |I (a) \ I˜(a)| ≤ K1−χ /2. Then summing over a ∈ [[1, M0]], with M0 ∼ |J |/K, we
get

Eft μG

1 |J | j∈J Gj,n

(6.44)

=1

Eft μG

|J | a : I (a)∩J =∅

Gj,n x(a) fty(a) dμyG(a) 1 G(a)
j ∈I˜(a)∩J

+ O N −c + O K−χ /2 ,

on . Since for each choice of the index a the term in the expectation on the right-hand side of (6.44) can be dealt with as in the case |J | ≤ 2K˜ + 1 above, this
completes the proof of (6.41) for general J .

We can now give the proof of Theorem 6.1.

PROOF OF THEOREM 6.1. Let α > 0. We first choose the constants a ∈ (0, 1/2), c ∈ (0, 1) and ε > 0, and the parameter K ∈ [[N σ , N 1/4]] appropriately:
let δ > 0 be a small constant. Then we set a ≡ 1/2 − δ, c ≡ δ/4, K ≡ N δ/4, ε ≡ δ,

2398

LEE, SCHNELLI, STETLER AND YAU

σ = δ/8. Note first that for this choice of K condition (6.4) is satisfied. Second, for sufficiently small δ > 0, we observe that

KN 2cN −aτ −1 = N 3δ/4N −aτ −1 ≤ Kχ ,

holds, for example, for τ ≥ N δN −a and χ > 0 (with χ < χ0). Thus (6.33) is satis-
fied with the above choices. Hence, for t ≥ N 2δτ , Lemma 6.9 yields, for some b > 0,

1 |J |

j

∈J

Gj,n(x)(ft

dμG

−

ψt

dμG)

≤

CO

K −b

+

CO

N −c

+

CO K−χ

/2

+

CO

√ KNcN

−aτ

−1,

for N sufficiently large on . Thus, choosing τ ≥ N δN −a, there is a constant f > 0 such that (6.3) holds. This completes the proof of Theorem 6.1.

Next, we sketch the proof of Theorem 6.2.

PROOF OF THEOREM 6.2. The proof of Theorem 6.2 is almost identical to

the proof of Theorem 6.1. In fact, it suffices to establish Lemma 6.8 with μG

replacing ψt μG on the left-hand side of (6.34). This can be accomplished by applying Theorem 6.5 with μG instead of ψt μG: let y˜ ∈ R∗μG(χ 2σ/2, α/2), and let y ∈ G(χ 2σ/2, α/2). Using the arguments of Proposition 5.2 in [31], we can rescale

μG such that (6.26) and (6.27) are satisfied for y and y˜. It is also straightforward

to check that the external potentials leading Kχ -regular. By Lemma 5.1 of [31] we obtain

to

μyG˜ ,

y˜

∈

R∗μG(χ 2σ/2, α/2),

are

EμyG˜ xk − αk ≤ CKχ N −1.

Hence, using estimate (6.35), we conclude that assumption (6.28) is also satisfied. Thus Theorem 6.5 yields

(6.45)

GL+p,n(x)ψty dμyG − GL+p,n,sc(x) dμyG˜ ≤ CO K−b,

for N sufficiently large on . We refer to the proof of Proposition 5.2 in [31] for
more details. Since R∗μG(χ 2σ/2, α/2) has exponentially high probability under μG, we can
integrate over y˜ to find

GL+p,n(x)ψty dμyG − GL+p,n,sc(x) dμG ≤ CO K−b,

for N sufficiently large on . The proof of Theorem 6.2 is now completed in the same way as the proof of
Theorem 6.1.

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2399

7. From gap statistics to correlation functions. In this section, we translate our results on the averaged local gap statistics into results on averaged correlation functions. Since this procedure is fairly standard (see, e.g., [29]), we refrain from stating all proofs in detail. We first need to slightly generalize the setup of Section 6.
Fix n ∈ N, let O be an n-particle observable and consider an array of increasing positive integers,

(7.1)

m = (m1, m2, . . . , mn) ∈ Nn.

Let α > 0. We define for j ∈ [[αN, (1 − α)N ]] and t ≥ 0 an observable Gj,m,t ≡ Gj,m by

(7.2)

Gj,m(x) := O Nρj (xj+m1 − xj ), Nρj (xj+m2 − xj ), . . . , Nρj (xj+mn − xj ) ,

where ρj ≡ ρfc(t, γj (t)) denotes the density of the measure ρfc(t) at the classical location of the j th particle, γj (t), with respect to the measure ρfc(t). We set Gj,m = 0 if j + mn ≥ (1 − α)N . Similarly, we define Gj,m,sc by replacing ρj by the density of the standard semicircle law at the classical locations of the j th parti-
cle with respect to the semicircle law; cf. (6.2). The following theorem generalizes
Theorem 6.2.

THEOREM 7.1. Let n ∈ N be fixed, and let O be an n-particle observable.
Fix small constants α, δ > 0, and consider an interval of consecutive integers
J ⊂ [[αN, (1 − α)N]] in the bulk. Then there are constants f, δ > 0 such that the following holds. Let m ∈ Nn be an array of increasing integers [see (7.1)] such that mn ≤ N δ , and consider the observable Gj,m, respectively, Gj,m,sc; see (7.2). Assume that t ≥ N −1/2+δ, then

1 |J |

j ∈J

Gj,m(x)ft

(x)

dμG(x)

−

1 |J | j∈J Gj,m,sc(x) dμG(x)

≤ CO N −f,

for N sufficiently large on . The constant CO depends on α and O, and the constants f and δ depend on α and δ.

Theorem 7.1 is proven in the same way as Theorem 6.2. We remark that δ

is chosen such that N δ K; that is, mn is much smaller than the size of the

interval IL,K .

For n ≥ 1, define the n-point correlation function,

N ft

,n,

by

N ft

,n

(x1

,

.

.

.

,

xn

)

:=

RN−n (ft μG)# dxn+1 · · · dxN ,

where (ft μG)# denote the symmetrized versions of ft μG. Similarly, we denote by

GN,n(x1, . . . , xn) := RN−n μ#G dxn+1 · · · dxN ,

2400

LEE, SCHNELLI, STETLER AND YAU

the n-point correlation functions of the Gaussian ensembles; see (4.13) with U ≡ 0.
Recall that we denote by L±(t), respectively, L±(t), the endpoints of the support of the measure ρfc(t), respectively, the measure ρfc(t). Recall that the two densities ft and ψt are both conditioned on V ; that is, the entries (vi) of V are considered fixed. We have the following result on the averaged correlation func-
tions of ft μG and ψt μG.

THEOREM 7.2. Fix n ∈ N, and choose an n-particle observable O. Fix a small δ > 0, and let t ≥ N −1/2+δ. Let α˜ > 0 be a small constant, and consider
two energies E ∈ [L−(t) + α˜ , L+(t) − α˜ ] and E ∈ [−2 + α˜ , 2 − α˜ ]. Then we have, for any ε > 0 and for b ≡ bN satisfying α˜ /2 ≥ bN > 0,

dα1 · · · dαnO(α1, . . . , αn)
Rn

(7.3)

×

E+b dx

1

E−b 2b [ρfc(t, E)]n

N ft ,n

x + α1 , . . . , x + αn

Nρfc(t, E)

Nρfc(t, E)

−

E +b dx 1 E −b 2b [ρsc(E )]n

N G,n

x + α1 , . . . , x + αn

Nρsc(E )

Nρsc(E )

≤ CO N 2ε b−1N −1+ε + N −f + N −cα0 ,

for N sufficiently large on . Here a is the constant in the rigidity estimate (5.6),
and f is the constant in Theorem 7.1. Moreover, ρfc(t, E) stands for the density of the (N -independent) measure ρfc(t) at the energy E. The constant CO depends on O and α˜ . Further, α0 is the constant appearing in Assumption 2.2. The constant c depends on the measure ν.

Theorem 7.2 follows from Theorem 6.2. This is an application of Section 7
in [29]. The validity of Assumption IV in [29] is a direct consequence of the
local law in Theorem 3.3. Further, we remark that the parameter bN in Theorem 7.2 and the interval of consecutive integers J in Theorem 7.1 are related by J = {i : γi(t) ∈ [E − bN , E + bN ]}, where γi(t) are the classical locations with respect to the measure ρfc(t). This explains, up to minor technicalities, bN N −1. Then Section 7 of [29] yields (7.3) formulated in terms of ρfc(t) instead of ρfc(t). Using (3.22) and the smoothness of O, we can replace ρfc by ρfc at the expense of an error of size CO N −cα0 . This eventually gives (7.3) with ρfc(t).

8. Proofs of main results. Theorem 7.2 shows that the averaged local correlation functions of ensembles of the form
Ht = e−(t−t0)/2V + e−t/2W + 1 − e−t 1/2W ,

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2401

with some small t0 ≥ 0, and with W a GUE/GOE matrix independent of W and V , can be compared with the averaged local correlation functions of the GUE, respectively, GOE, for times satisfying t N−1/2. In this section, we explain how this
can be used to prove the universality at time t = 0.

8.1. Green function comparison theorem. We start with a Green function comparison theorem. Assume that we are given two complex Hermitian or real symmetric Wigner matrices, X and Y , both satisfying the assumptions in Definition 2.1. Let V be a real random or deterministic diagonal matrix satisfying Assumptions 2.3 and 2.2. Consider the deformed Wigner matrices

(8.1)

H X := V + X, H Y := V + Y,

of size N . The main theorem of this subsection, Theorem 8.2, states that the correlation functions of the two matrices H X and H Y , when conditioned on V , are
identical on scale 1/N provided that the first four moments of X and Y almost
match. Theorem 8.2 is a direct consequence of the Green function comparison
Theorem 8.1. Denote the Green functions of H X, H Y , respectively, by

GX (z)

:=

H

1 X−

z

,

GY

(z)

:=

H

1 Y−

z

(z ∈ C \ R),

and set mXN (z) := N −1 Tr GX(z), mYN (z) := N −1 Tr GY (z). From Theorem 3.3, we know that, for all z ∈ DL [see (3.9)], with L ≥ 40ξ ,

(8.2)

mXN (z) − mfc(z)

≤

(ϕN

)cξ

1 Nη

and

(8.3)

GXij (z) − δij gi (z) ≤ (ϕN )cξ

Im mfc(z) + 1 , Nη Nη

with (ξ, υ)-high probability on for some υ > 0 and c > 0, where

gi (z)

:=

vi

−

z

1 −

mfc(z)

(z ∈ C \ R).

Here, mfc, is the Stieltjes transform of the measure ρfc, which agrees with ρfϑc for the choice ϑ = 1 and with ρfc(t) for the choice t = t0. The identical estimates hold true when X is replaced by Y .
Recall that we denote by L± the endpoints of the support of ρfc, and that we denote by κE ≡ κ the distance of E ∈ [L−, L+] to the endpoints L±. Adapting the Green function theorem of [32] we obtain the following theorem.

2402

LEE, SCHNELLI, STETLER AND YAU

THEOREM 8.1 (Green function comparison theorem). Assume that X and Y satisfy the assumptions in Definition 2.1, and let V satisfy Assumptions 2.2 and 2.3. Assume further that the first two moments of X = (xij ) and Y = (yij ) agree and that the third and forth moments satisfy

(8.4)

Ex¯ipj xi3j−p − Ey¯ipj yi3j−p ≤ N −δ−2

p ∈ [[0, 3]] ,

respectively,

(8.5)

Ex¯iqj xi4j−q − Ey¯iqj yi4j−q ≤ N −δ

q ∈ [[0, 4]] ,

for some given δ > 0. Let ε > 0 be arbitrary, and let N −1−ε ≤ η ≤ N −1. Fix N -independent integers
k1, . . . , kn and energies Ej1, . . . , Ejkj , j = 1, . . . , n, with κ > α˜ for all Ejk with
some fixed α˜ > 0. Define zjk := Ejk ± iη, with the sign arbitrarily chosen. Suppose that F is a smooth function such that for any multi-index σ = (σ1, . . . , σn), with
1 ≤ |σ | ≤ 5, and any ε > 0 sufficiently small, there is a C0 > 0 such that

max ∂σ F (x1, . . . , xn) : max |xj | ≤ N ε ≤ N C0ε ,
j
max ∂σ F (x1, . . . , xn) : max |xj | ≤ N 2 ≤ N C0,
j

for some C0. Then there exists a constant C1, depending on m km, C0 and the constants
in (2.3), such that for any η with N −1−ε ≤ η ≤ N −1,

(8.6)

EF

1 N k1

k1
Tr GX
j =1

zj1

,...,

1 N kn

kn
Tr GX
j =1

zjn

− EF

1 N k1

k1
Tr GY
j =1

zj1

,...,

1 N kn

kn
Tr GY
j =1

zjn

≤ C1N −1/2+C1ε + C1N −1/2+δ+C1ε,

for N sufficiently large on .

Theorem 8.1 is proven in the same way as Theorem 2.3 in [33] with the fol-
lowing modifications. Fix some labeling of {(i, j ) : 1 ≤ i ≤ j ≤ N} by [[1, γ (N)]],
with γ (N ) := N (N + 1)/2, and write the γ th element of this labeling as (iγ , jγ ). Starting with W (0) ≡ X, inductively define W (γ ) by replacing the (iγ , jγ ), (jγ , iγ ) entries of W (γ −1) by the corresponding entries of Y . Moreover set H (γ ) := V + W (γ ). Thus we have H (0) = H X, H (γ (N)) = H Y , and H (γ ) − H (γ −1) is zero

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2403

in all but two entries for every γ . In short, we use a Lindeberg-type replacement strategy: we successively replace the entries of the matrix X by entries of the matrix Y . Note, however, that the entries of the matrix V are not changed.
The main technical input in the proof of Theorem 2.3 in [32] is estimate (2.21) in that publication. For the case at hand the corresponding estimate reads as follows: let ξ satisfy (2.20). Then, for all δ > 0, and any N −1/2 y ≥ N −1+δ, we have

(8.7)

P max max sup
γ ≤γ (N) k E : κE ≥α˜
≤ e−υ(ϕN )ξ ,

1 H (γ ) − E − iy

≥ N 2δ
kk

on for N sufficiently large, where υ > 0 depends only on α˜ , δ and the constants in (2.3). Estimate (8.7) follows easily from the local law in (8.3), the stability bound (3.21) and Lemma 3.6. The rest of the proof of Theorem 8.1 is identical to the proof in [32]. (The matching conditions in (8.4) are weaker than in [32], but the proof carries over without any changes.)
Lindeberg’s replacement method was applied in random matrix theory in [15] to compare traces of Green functions. This idea was also used in [56] in the proof of the “four moment theorem” that compares individual eigenvalue distributions. The four-moment matching conditions (8.4) and (8.5) appeared first in [56] with δ = 0. The “Green function comparison theorem” of [32] compares Green functions at fixed energies. Since the approach in [56] requires additional difficult estimates due to singularities from neighboring eigenvalues, we follow the method of [32], where difficulties stemming from such resonances are absent. For deformed Wigner matrices with deterministic potential the approach of [56] was recently followed in [45] where a “four moment theorem” was established. It allows one to compare local correlation functions of the matrices V + W and V + W for fixed V , where W and W are real symmetric or complex Hermitian Wigner matrices, provided that the moments of the off-diagonal entries of W and W match to fourth order.
The Green function comparison theorem leads directly to the equivalence of local statistics for the matrices H Y and H X.

THEOREM 8.2. Assume that X, Y are two complex Hermitian or two real

symmetric Wigner matrices satisfying assumptions in Definition 2.1. Assume fur-

ther that X and Y satisfy the matching conditions (8.4) and (8.5), for some δ > 0.

Let V be a deterministic real diagonal matrix satisfying the Assumptions 2.2

and 2.3. Denote by

N H

X

,n

,

N H Y ,n

the

n-point

correlation

functions

of

the

eigen-

values with respect to the probability laws of the matrices H X, H Y , respectively.

Then, for any energy E in the interior of the support of ρfc and any n-particle

2404

LEE, SCHNELLI, STETLER AND YAU

observable O, we have

lim
N →∞

dα1 · · · dαnO(α1, . . . , αn)
Rk

×

N H X,n

E + α1 , . . . , E + αn NN

= 0,

−

N H Y ,n

E + α1 , . . . , E + αn NN

for any fixed n ∈ N.

Notice that this comparison theorem holds for any fixed energy E in the bulk.
The proof of [32] applies almost verbatim. The only technical input in the proof is the local law for mXN , respectively, mYN , on scales η ∼ N −1+ε, which we have established in Theorem 3.3; see also (8.3).

8.2. Proof of Theorem 2.5. In the remaining subsections, 8.2 and 8.3, we complete the proofs of our main results in Theorems 2.5 and 2.6. The proofs for deterministic and random V differ slightly. We start with the case of deterministic V in this subsection; the random case is treated in Section 8.3.

PROOF OF THEOREM 2.5. Assume that W = (wij ) is a complex Hermitian or a real symmetric Wigner matrix satisfying the assumptions in Definition 2.1. Let
V = diag(vi) be a deterministic real diagonal matrix satisfying Assumptions 2.2 and 2.3. (Note that the event then has full probability.) Set H = (hij ) = V + W . Let E ∈ R be inside the support of ρfc. Note that by Lemma 3.6, E is also contained in the support of ρfc, for N sufficiently large. (Here we have ρfc = ρfϑc=1 and similarly for ρfc.) Fix δ > 0, and set t ≡ N −1/2+δ . We first claim that there exists an auxiliary complex Hermitian or real symmetric Wigner matrix, U = (uij ), satisfying the assumptions in Definition 2.1 such that the following holds: set

(8.8)

Y := e−t/2U + 1 − et/2 W ,

where W is a GUE/GOE matrix independent of W . Then the moments of the entries of Y satisfy

(8.9)

Ey¯ipj yi3j−p = Ew¯ ipj wi3j−p,

Ey¯iqj yi4j−q − Ew¯ iqj wi4j−q ≤ Ct,

for p ∈ [[0, 3]], q ∈ [[0, 4]], where (wij ) are the entries of the Wigner matrix W . Assuming the existence of such a Wigner matrix U , we choose t0 ≡ t and set

Ht := e−(t−t0)/2V + e−t/2U + 1 − e−t 1/2W

= V + e−t/2U + 1 − e−t 1/2W .

Then the matrices Ht and H = V + W satisfy the matching conditions (8.4) and (8.5) of Theorem 8.1 (with, say, δ = 1/4 − 2δ ). This follows from (8.9). Thus

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2405

Theorem 8.2 implies that the correlation functions of Ht and H agree in the limit of large N , that is,

lim
N →∞

dα1 · · · dαnO(α1, . . . , αn)
Rn

(8.10)

×

1 2b

E+b dx E−b [ρfc(E)]n

N Ht ,n

x + α1 , . . . , x + αn

ρfc(E)N

ρfc(E)N

−1 2b

E+b dx E−b [ρfc(E)]n

N H ,n

x + α1 , . . . , x + αn

ρfc(E)N

ρfc(E)N

= 0,

where (

N H

,n)

denote

the

correlation

functions

of

H

=V

+W

and where (

N Ht

,n)

denote the correlation functions of Ht . [In fact, (8.10) holds even without the av-

erages in the energy around E.]

On the other hand, for small δ > 0, Theorem 7.2 assures that the local corre-

lation functions of the matrix Ht agree with the correlation functions of the GUE (resp., GOE), when averaged over an interval of size b, with 1 b ≥ N −δ; that is,

for any E with |E | < 2,

lim
N →∞

dα1 · · · dαnO(α1, . . . , αn)
Rn

(8.11)

×

1 2b

E+b dx E−b [ρfc(E)]n

N Ht ,n

x + α1 , . . . , x + αn

ρfc(E)N

ρfc(E)N

1 − [ρsc(E )]n

N G,n

E

+ α1 , . . . , E + αn

ρsc(E )N

ρsc(E )N

= 0,

where ( GN,n) denote the correlations functions of the GUE, respectively, GOE. Combining (8.10) and (8.11), we get (2.15).
Thus to complete the proof we need to show the existence of a Wigner ma-
trix U with the properties described above. For a real random variables ζ , denote by mk(ζ ) = Eζ k, k ∈ N, its moments.

LEMMA 8.3 (Lemma 6.5 in [32]). Let m3 and m4 be two real numbers such that
m4 − m22 − 1 ≥ 0, m4 ≤ C1,
for some constant C1. Let ζG be a Gaussian random variable with mean 0 and variance 1. Then for any sufficient small γ > 0, depending on C1, there exists a real random variable ζγ with subexponential decay and independent of ζG, such that the first three moments of
ζ := (1 − γ )1/2ζγ + γ 1/2ζG

2406

LEE, SCHNELLI, STETLER AND YAU

are m1(ζ ) = 0, m2(ζ ) = 1, m3(ζ ) = m3, and the forth moment m4(ζ ) satisfies m4 ζ − m4 ≤ Cγ ,
for some C depending on C1.

Since the real and imaginary parts of W are independent, it is sufficient to match them individually; that is, we apply Lemma 8.3 separately to the real and imaginary parts of (wij ). This completes the proof of Theorem 2.5 for deterministic V .

8.3. Proof of Theorem 2.6. Next, we prove Theorem 2.6. Assume that W =

(wij ) is a complex Hermitian or a real symmetric Wigner matrix satisfying the assumption in Definition 2.1. Let V = diag(vi) be a random real diagonal matrix satisfying Assumptions 2.2 and 2.3. Denote by f˜t μG the distribution of the eigen-
values of the matrix

Ht := V + e−t/2W + 1 − et 1/2W

(t ≥ 0),

where W is a GUE/GOE matrix independent of V and W . Let EV stand for the

expectation with respect to the law of the entries (vi) of V . Recall the definition of the event in Definition 3.3. Following the notation of Section 5, ft μG ≡ ftV μG denotes the density conditioned on V . For an n-particle observable O and for Gj,m
as in (7.2), we may write

1 |J |

j ∈J

Gj,m(x)f˜t

(x)

dμG(x)

= EV

1 |J |

j

∈J

Gj,m(x)ftV

(x)

dμG(x)

1(

)

+ O N −t ,

where t > 0 is the constant in (2.12) of the Assumptions in 2.3. Here we use the definition of . Since (vi) are i.i.d., (3.3) holds with exponentially high probability. Estimate (3.4) holds with probability large than 1−N −t by Assumption 2.3. Hence PV ( c) ≤ cN −t, for some c > 0 and N sufficiently large.
Using Theorem 7.1, we find that

(8.12)

1 |J

|

j

∈J

Gj,m(x)f˜t

(x)

dμG(x)

=

1 |J |

j ∈J

Gj,m,sc(x) dμG(x)

+O

N −f

+ O N −t ,

where we use once more the estimates on the event . Here f > 0 is the constant appearing in Theorem 7.1.
To establish the equivalent result to Theorem 7.2, we need a local deformed semicircle law for the setting when the entries (vi) of V are not fixed. Recall that we denote by mfc the Stieltjes transform of the deformed semicircle law ρfc = ρfϑc=1.

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2407

LEMMA 8.4 (Theorems 2.10 and 2.21 in [39]). Let W be a complex Hermitian
or a real symmetric Wigner matrix satisfying the assumptions in Definition 2.1.
Let V be a random real diagonal matrix satisfying Assumptions 2.2 and 2.3. Set H := V + W , G(z) := (H − z)−1 and mN (z) := N −1 Tr G(z), (z ∈ C+). Let ξ = A0 log log N/2; see (2.20). Then there exists υ > 0 and c [both depending on the constants in (2.3), the constants A0, E0 in (3.9) and the measure ν], such that for L ≥ 40ξ , we have

(8.13)

mN (z) − mfc(z) ≤ (ϕN )cξ

min

1 N 1/4

,

√1 κE +

η

√1 N

+1 , Nη

and

(8.14)

Gij (z) − δij gi(z) ≤ (ϕN )cξ

Im mfc(z) + 1 , Nη Nη

i, j ∈ [[1, N]], with (ξ, υ)-high probability, for all z = E + iη ∈ DL; see (3.9). Here, we have set

1 gi(z) := vi − z − mfc(z)

z ∈ C+, i ∈ [[1, N ]] .

Moreover, fixing α > 0, there is c1 [depending on the constants in (2.3), the constants A0, E0 in (3.9), the measure ν and α], such that

(8.15)

|λi − γi| ≤ (ϕN )c1ξ √1 , N

with (ξ, υ)-high probability, for all i ∈ [[αN, (1 − α)N ]]. Here (λi) denote the eigenvalues of H = V + W , and (γi) are their classical locations with respect the deformed semicircle law ρfc.

Using the local law in Lemma 8.4, we obtain from (8.12) equivalent results to Theorem 7.2.

THEOREM 8.5. Fix n ∈ N, and consider an n-particle observable O. Fix δ > 0, and let t ≥ N −1/4+δ. Let α˜ > 0 be a small constant, and consider two ener-
gies E ∈ [L−(t) + α˜ , L+(t) − α˜ ] and E ∈ [−2 + α˜ , 2 − α˜ ]. Then, for any ε > 0 and for b ≡ bN satisfying α˜ /2 ≥ bN > 0, we have

dα1 · · · dαnO(α1, . . . , αn)
Rn

×

E+b dx

1

E−b 2b [ρfc(t, E)]n

N f˜t ,n

x + α1 , . . . , x + αn

Nρfc(t, E)

Nρfc(t, E)

−

E +b dx 1 E −b 2b [ρsc(E )]n

N G,n

x + α1 , . . . , x + αn

Nρsc(E )

Nρsc(E )

≤ CO N 2ε b−1N −1/2+ε + N −f + N −t + N −1/4 ,

2408

LEE, SCHNELLI, STETLER AND YAU

for N sufficiently large. Here, f > 0 is the constant in Theorem 7.1. Moreover,
ρfc(E) stands for the density of the (N -independent) measure ρfc at the energy E. The constant CO depends on O, α˜ and the measure ν. The constant f depends on δ and α˜ .

The proof of Theorem 8.5 is an application of Section 7 in [29]. The validity
of Assumption IV in [29] is a direct consequence of the local law in Lemma 8.4.
Here and also below, we use that the local laws of Lemma 8.4 are only used on very small scales η ∼ N −1+ε in the bulk. For such small η the first error term
in (8.13) is negligible compared to the second error term. Also note that the first
term on the right-hand side of the estimate in Theorem 8.5 is bigger than the cor-
responding term in (7.3). This is due to the weaker rigidity bounds in case V is random; see (8.15). We therefore have to impose that b N −1/2 in order to have
a vanishing error term in the limit of large N . Finally, we mention that the error term CO N 2εN −1/4 stems from replacing ρfc(t, E) by ρfc(t, E); see the comment below Theorem 7.2.

PROOF OF THEOREM 2.6. The proof Theorem 2.6 follows now along the lines of the proof of Theorem 2.5. First, we check that the Green function comparison Theorem 8.1 holds true for H X = V + X, respectively, H Y = V + Y with random V . This is indeed the case, since the only input we used is estimate (8.7), which also holds for random V by the local laws in Lemma 8.4 and the stability estimate (3.21). Note that we are using that bound (8.7) is only required on scales η N−1/2. Similarly, we can establish Theorem 8.2 for random V using the Green function comparison theorem for random V , the local laws in Lemma 8.4 and the stability estimate (3.21). Finally, we note that the construction of the matrix U and Y [see (8.8)] and the moment matching in (8.9) do not involve V . We can thus complete the proof of Theorem 2.6 in the same way as the proof of Theorem 2.5.

9. Edge universality for deformed Wigner matrices. In this section we prove Theorem 2.10. Its proof is a combination of Corollary 5.4 (bounds on the global Dirichlet form) and the method of [12]. In fact, the proof of the edge universality is very similar to the proof of the bulk universality: we first establish the edge universality for our model with a small Gaussian component (cf. Section 6 for the bulk), and then remove the small Gaussian component using Green function comparison and a moment matching; cf. Section 8 for the bulk.

9.1. Edge universality with a small Gaussian component. We mainly follow the exposition in Section 3 of [12]. We consider the local statistics at the lower edge; the upper edge is treated in exactly the same way.

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2409

9.1.1. Preliminaries. Recall the definition of the β-ensemble μU in (4.2) for a given potential U that is C4 and “regular.” To study the local statistics at the lower edge, we introduce two auxiliary measures, σ and σˇ , on (N) as follows.
By a shift and a rescaling, we can assume that the equilibrium density, U , of μU is supported on [0, A+], for some A+ > 0. Fix a small ε0 > 0, and set

(9.1)

σ (dλ) := 1 e−βNHσ (λ) dλ, Zσ

with

(9.2)

Hσ (λ)

=

H(λ) +

2 N

N i=1

N 2/3−ε0 λi ,

(x) := (x + 1)21(x < −1),

where H is given in (4.3) and where Zσ ≡ Zσ (β) is a normalization. Similarly, we introduce

(9.3)

σˇ (dλ) := 1 e−βNHσˇ (λ) dλ, Zσˇ

with

(9.4)

1N Hσˇ (λ) = H(λ) + N i=1

N 2/3−ε0 λi ,

with Zσˇ ≡ Zσˇ (β) a normalization. The potential is added to avoid that the (xi) deviate too far to the left, yet its influence on the local statistics at the edge is
negligible; see Lemma 4.1 in [12]. Below, we choose β = 1, 2 depending on the
symmetry class of our original matrix.
Following Section 3 of [12], we choose a small δ > ε0 and an integer K such that K ∈ [[N δ, N 1−δ]]. Denote by I = [[1, K]] the set of the first K indices. For λ ∈ (N), we write

(9.5)

(λ1, λ2, . . . , λN ) = (x1, . . . , xK , yK+1, . . . , yN ),

and (9.6)

x = (x1, . . . , xK ) ∈ (K),

y = (yK+1, . . . , yN ) ∈ (N−K);

cf. (6.5) and (6.6). We further denote I localized measures μyU , σ y and σˇ y as

:= (−∞, yK+1]. For fixed y, we define the in Section 6.2. (For simplicity of notation,

we do not indicate the U and ε0 dependences in the measures σ , σˇ .)

We introduce the set of “good” boundary conditions

(9.7) R(ε0) ≡ R := y ∈ (N−K) : |yk − γk| ≤ N −2/3+ε0 kˇ, k ∈/ I ,

with kˇ = min{k, N − k}, where (γk) denote the classical locations with respect
to the equilibrium density. With our choices of δ and ε0, we have yK − y1 ∼ (K/N )2/3.

2410

LEE, SCHNELLI, STETLER AND YAU

9.1.2. Comparison of the local measures at the edge. Fix t > 0. Recall that
we denote by ft μG the distribution of λ(t) under the flow generated by (5.3). As in Section 6, we fix (vi), and condition of the event ; see Definition 3.1. Also recall from (4.21) the definition of the time dependent reference β-ensemble
ψt μG, whose equilibrium density is fc(t). By a simple shift and a scaling, we may assume, for fixed t, that supp fc = [0, L+(t)] and that

(9.8)

fc(t, x) =

1√ x
π

1 + O(x)

,

as x 0. This can easily be checked from the proofs of the Lemmas A.1 and 3.6

in the ftyμyG

Appendix. For y ∈ R, we then introduce the in the obvious way. For technical reasons, we

localized measures ψtyμyG and also use the measures σ and σˇ ,

with the choice U = U (t). (The Hamiltonians of the measures ψt μG and σ , σˇ ,

agree up to the confining potential .)

In a first step, we compare the statistics of ψtyμyG and σ y. This is the analogue

result to Proposition 6.4 above, respectively, to Lemma 5.4 in [12].

LEMMA 9.1. Let 0 < a < 1/2. Fix small constants δ > ε0 > 0. Let K ∈ [[N δ, N 1−δ]], and let O be an n-particle observable. Let ε > 0, and choose τ satisfying 1 τ > N −2a. Then, for any t ≥ N ε τ and any constant c ∈ (0, 1), there is
a set of configurations G(ε0) ≡ G ⊂ R, with

(9.9)

Pft μG (G) ≥ 1 − N −c , 2

such that

(9.10)

O(x) fty(x)μyG(dx) − σ y(dx) ≤ CO K1/6N 1/3N c−aτ −1,

t ≥ N ε τ , for N sufficiently large on . Moreover, there is υ > 0, such that

(9.11)

PftyμyG xk − γk(t ) < N −1+ε0 , k ∈ I ≥ 1 − e−υ(ϕN )ξ ,

t ≥ N ε τ , for N sufficiently large on , with ξ = A0 log log N/2; see (2.20).

PROOF. We follow the proof of Proposition 6.4 with some modifications. First, introduce the density qt by demanding

qt σ = ft μG.

Then we note that, at the lower edge,

1 N

k∈I c

(x

1 − γk(t))2

≥

cN 1/3/K1/3,

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2411

for for

x≥ Hσy .

−N −2/3+ε0 and y ∈ R. We thus have ∇x2Hσy Hence the logarithmic Sobolev inequality

(x)

≥

cN

1/3/K

1/3;

cf.

(6.10)

Sσ y

qty

K 1/3 ≤ C N 1/3 Dσ y

qty ,

with

the

Dirichlet

form

Dσ y (f

)

=

1 βN

i∈I |∂if (x)|2σ y(dx) holds. To bound the

Dirichlet form, we proceed as in (6.24),

Eσ Dσ y

qty

≤

Dσ

√ ( qt

)

N
≤ 2Dψt μG ( gt ) + CN 4/3 Eψt μG
i=1

≤

N 1−2a C τ2

+ e−Nc ,

N 2/3−ε0 xi 2

for some c > 0, with gt = ft /ψt , where we used the definitions of Dσ , Dψt μG to get the second line. The third line follows from Corollary 5.4 and Lemma 5.5.
To complete the proof, we now follow mutatis mutandis the proof of Proposi-
tion 6.4. We leave the details aside.

Eventually, we are going to apply Theorem 3.3 of [12], which shows that the statistics of σ y are universal for most boundary conditions y. In order to apply it,
we need the analogue of Lemma 6.7 above.

LEMMA 9.2. Under the assumptions of Proposition 9.1 the following holds. Let y ∈ G. Then, assuming that

(9.12)

K1/3N −1/3N 2c−aτ −1 ≤ N ε0 K−1/3N −2/3,

we get, for all k ∈ I ,

(9.13)

EftyμyG xk − Eσ y xk ≤ CN ε0 K−1/3N −2/3,

for N sufficiently large on .

PROOF. Replacing the constant τK = CK/N in the logarithmic Sobolev inequality 6.23 by CK1/3/N 1/3, we can copy the proof of Lemma 6.7 (see also Lemma 5.5 in [12]) almost word by word.

From (9.13), we immediately get, for y ∈ G, the estimate

(9.14)

Eσ y xk − γk(t ) ≤ CN ε0 N −2/3k−1/3,

provided that (9.12) holds.

2412

LEE, SCHNELLI, STETLER AND YAU

9.1.3. Universality of the localized measures at the edge. In this subsection, we establish the following result.

LEMMA 9.3. Fix an integer n > 0. Then for any 1/4 > κ the following holds on the event . For any δ > 0, there is a constant f > 0 such that, for t ≥ N −δ and for ⊂ [[1, N κ]] with | | = n,

Eft μG O ct N 2/3j 1/3 λj − γj (t ) j∈

(9.15)

− EμG O N 2/3j 1/3(λj − γj ) j∈ ≤ CN −f,

where ct depends only on fc(t). Here, (γj ) denote the classical locations with respect the measure t , and (γj ) denote the classical locations with respect the semicircle law sc.

PROOF. We follow the proof of Lemma 5.1 in [12]. We consider the case
n = 1 only; the general case is proved in the same way. By a shift and a scaling, we may assume that ct = 1 [see (9.8)], and we may replace γj (t) by the γj . [Here, we implicitly use that we fixed (vi) and conditioned on the event .]
We will need two modifications of the set R(ε0) of “good” boundary conditions. Let σ , σˇ be given by (9.1), respectively (9.3) (with a generic potential U ). Then
set

R∗(ε0) := y ∈ R(ε0) : ∀k ∈ I, Eσ y xk − γk ≤ N −2/3+ε0 k−1/3,

(9.16)

Pσˇ y x1 ≥ γ1 − N −2/3+ε0 ≥ 1/2 .

We further need the set

(9.17) R#(ε0) := y ∈ R(ε0/3) : |yK+1 − yK+2| ≥ N −2/3−ε0 K−1/3 .
While the set R∗(ε0) incorporates rigidity estimates in the sense that γk is a good approximation in expectation to xk and that x1 is not too much on the left, the set R#(ε0) incorporates a level repulsion estimate. It has no counterpart in Section 6 above.
We now choose a = 1/2 − δ , c = δ /2 and τ = N −δ , for some small 1/12 > δ > 0. With this choice, we have for K ≤ N 1/4−6δ , that

(9.18)

K1/6N 1/3N c−aτ −1 ≤ N −ε0 ,

respectively, (9.19)

K1/3N −1/3N 2c−aτ −1 ≤ N ε0 K−1/3N −2/3,

for a small ε0 > 0 (with δ > ε0).

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2413

Then, from Lemma 9.1 we have, for y ∈ G,

(9.20)

O N 2/3j (xj − γj ) fty dμyG − dσ y ≤ CO N −χ

(j ∈ ),

for some χ > 0. Here, the measure σ is given by (9.1) with the potential U (t).
Let σ˜ denote the measure given by (9.1) with the potential U ≡ 0. For y˜ ∈
R(ε0) (where the classical locations are taken with respect the semicircle law), we introduce the localized measure σ˜ y˜ . We now apply Theorem 3.3 of [12]: for y ∈ R#(ε0) ∩ R∗(ε0), respectively, y˜ ∈ R#(ε0) ∩ R∗(ε0), we have

(9.21)

O N 2/3j (xj − γj ) dσ y − dσ˜ y˜ ≤ CO N −χ ,

for sufficiently large N , by choosing χ > 0 sufficiently small. From Lemma 4.1 of [12], we know that Pσ˜ (R#(ε0) ∩ R∗(ε0)) ≥ 1 − N −c, for some c > 0. We
further know from Lemma 4.1 of [12] that, for any bounded observable O, |Eσ˜ O − EμGO| ≤ CO exp(−N c), c > 0, where μG denotes the GUE/GOE. Thus, integrating out the boundary conditions y˜ and replacing σ˜ with μG, we get
from (9.20) and (9.21),

(9.22)

O N 2/3j (xj − γj ) fty dμyG − dμG ≤ CO N −χ ,

for sufficiently small χ > 0, where y ∈ G(ε0) ∩ R#(ε0) ∩ R∗(ε0). Once we have established that

(9.23)

Pft μG G(ε0) ∩ R#(ε0) ∩ R∗(ε0) ≥ 1 − N −c,

for some c > 0, we integrate out the boundary condition y in (9.22), and we get (9.15) for n = 1.
To prove (9.23) we follow the two steps of the proof of (5.23) in [12]. In a first step, one controls the probability of R#(ε0) using the rigidity estimates for ft μG (see Lemma 3.4), the level repulsion estimates for the measure σ y in Theorem 3.2
of [12], Lemma 9.1 and the condition (9.19). In a second step, one shows that G(ε0) ⊂ R∗(ε0). This follows from (9.2) and the arguments given in the proof of Lemma 5.1 of [12]. In this way (9.23) can be established; we leave the details to
the interested reader.

9.2. Removal of the Gaussian component. In this subsection we prove Theorem 2.10. We use the following version of the Green function comparison theorem at the edge. It is the counterpart to Theorem 8.1 above.

THEOREM 9.4. Suppose we have two Wigner matrices X and Y satisfying the
conditions in Definition 2.1. Set H X := V + X, H Y := V + Y ; see (8.1). Denote by PX, PY the probability distributions of X,Y . Then on the following holds true.

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LEE, SCHNELLI, STETLER AND YAU

For any ε > 0, there is δ > 0 [depending on ε and the constants C0, ϑ in (2.3)], such that

PX N 2/3(λ1 − γ1) ≤ s − N −ε − N −δ ≤ PY N 2/3(λ1 − γ1) ≤ s

≤ PX N 2/3(λ1 − γ1) ≤ s + N −ε + N −δ,

s ∈ R,

for N sufficiently large, where (γk) denote the classical locations of the mea-

sure fc ≡ fθc, the eigenvalues

with θ = 1. Analogous results λi1, λi2, . . . , λip , as long as |ip|

hold for ≤ Nε.

the

joint

distributions

of

Theorem 9.4 is proven exactly in the same way as Theorem 2.4 of [33] for the Wigner case V = 0. It suffices to note that the entries of V are fixed in Theorem 9.4 and that the only input needed in the proof are the local laws for the Green functions of H X and H Y , which have been established in Theorem 3.3 above.
Given Theorem 9.4, we now complete the proof of Theorem 2.10. Following
the arguments in Section 8.2, we construct an auxiliary Wigner matrix U such that
the first two moments of the matrix

(9.24)

Ht = V + e−t/2U + 1 − e−t 1/2W

with t = N −δ (δ > 0 as in Lemma 9.3, and W an independent GUE/GOE matrix) and the matrix H = V + W match. By Lemma 9.3 the edge statistics of Ht are universal. By Theorem 9.4 the eigenvalue statistics of Ht and H at the edge agree for large N . The existence of such U is assured by Lemma 8.3. We have thus
established that there is a small χ > 0 such that

(9.25)

Ef0μG O c0N 2/3j 1/3(λj − γj ) j∈ − EμG O N 2/3j 1/3(λj − γj ) j∈

≤ CN −χ ,

for N sufficiently large on , where μG is the GUE/GOE. Finally, we use Assumptions 2.2 and 2.3 as well as a simple moment bound to
average over ν (the empirical distribution of V ) in (9.25). This completes the proof of Theorem 2.10.

APPENDIX
In this Appendix we prove the auxiliary results used in Sections 3 and 4: Lemmas 3.5, 3.6 and 4.2. We start with a more extended version of Lemma 3.5. Recall from (3.1) that we denote = [0, 1 + ], = /10. Also recall the definition of the domain D of the spectral parameter z in (3.17).

LEMMA A.1. Let ν satisfy Assumption 2.3 for some > 0. Then the follow-

ing holds true for any ϑ ∈ . There are Lϑ−, Lϑ+ ∈ R, with Lϑ− < 0 < Lϑ+, such

that supp ρfϑc = [Lϑ−, Lϑ+], and there is a constant C > 1 such that, for all ϑ ∈ ,

(A.1)

C−1√κE

≤

ρfϑc(E)

≤

√ C κE

E ∈ Lϑ−, Lϑ+ ,

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2415

where κE denotes the distance of E to the endpoints of the support of ρfϑc, that is,

(A.2)

κE := min E − Lϑ− , E − Lϑ+ .

The Stieltjes transform, mϑfc, of ρfϑc has the following properties:

(1) for all z = E + iη ∈ C+,

⎧ ⎨

√ κ

+

η,

(A.3)

Im mϑfc(z) ∼ ⎩ √κη+ η ,

E ∈ [L−, L+], E ∈ [L−, L+]c;

(2) there exists a constant C > 1 such that for all z ∈ D and for all x ∈ Iν,

(A.4)

C−1 ≤ ϑx − z − mϑfc(z) ≤ C;

(3) there exists a constant C > 1 such that for all z ∈ D ,

(A.5)

C−1√κ + η ≤ 1 −

dν(v) (ϑv − z − mϑfc(z))2

√ ≤ C κ + η;

(4) there are constants C > 1 and c0 > 0 such for all z = E + iη ∈ D satisfying κE + η ≤ c0,

(A.6)

C−1 ≤

dν(v) (ϑv − z − mϑfc(z))3

≤ C;

moreover, there is C > 1, such that for all z ∈ D ,

(A.7)

dν(v) (ϑv − z − mϑfc(z))3 ≤ C.

The constants in statements (1)–(4) can be chosen uniformly in ϑ ∈

.

PROOF. We follow the proofs in [39, 51]. Let ϑ ∈ and let

. Set ζ = z + mϑfc(z),

(A.8)

F (ζ ) := ζ −

dν(v) ϑv − ζ

ζ ∈ C+ .

Then the functional equation (3.5) is equivalent to z = F (ζ ). As is argued in [51], a point E ∈ R is inside the support of the measure ρfϑc if and only if ζE = E + mϑfc(E) satisfies Im F (ζE) = 0 and Im ζE > 0. Accordingly, the endpoints of the support are characterized as the solutions of

(A.9)

H (ζ ) :=

dν(v) (ϑv − ζ )2

=

1

(ζ ∈ R).

Note that H (ζ ) is a continuous function outside ϑIν ≡ {x : x = ϑy, y ∈ Iν} which is decreasing as |ζ | increases. Since ϑ ∈ = [0, 1 + ], with = /10, we obtain from Assumption 2.3 that H (ζ ) ≥ 1 + /2, for all ζ ∈ ϑIν. It thus follows

2416

LEE, SCHNELLI, STETLER AND YAU

that there are only two solutions, ζ±ϑ ∈ R \ ϑIν , to H (ζ ) = 1, ζ ∈ R. In particular, ζ−ϑ < 0, ζ+ϑ > 0, and there is a constant g > 0, depending only on ν, such that

(A.10)

inf
ϑ∈

dist

ζ±ϑ , ϑ Iν

≥ g.

As argued in [39, 51], the set γ := {ζ ∈ C+ : Im F (ζ ) = 0, Im ζ > 0} is, for each
fixed ϑ ∈ , a finite curve in the upper half plane that is the graph of a continuous function which only connects to the real line at ζ±ϑ .
Since dist({ζ±ϑ }, ϑIν} ≥ g > 0, F (ζ ) is analytic in a neighborhood of ζ±ϑ . Thus for ζ in a neighborhood of ζ+ϑ , we may write

F (ζ ) = F ζ+ϑ + F

ζ+ϑ

ζ − ζ+ϑ

F +

(ζ+ϑ ) 2

ζ − ζ+ϑ

2+O

ζ − ζ+ϑ 3 .

Note that F (ζ+ϑ ) = 0 by the definition of ζ+ϑ . Moreover, we know that Im F (ζ ) = 0, for ζ in a real neighborhood of ζ+ϑ , but we also have Im F (ζ ) = 0, for ζ ∈ γ ∪ γ¯ . Thus F (ζ+ϑ ) = 0. We can therefore invert F (ζ ) = z in a neighborhood of ζ+ to obtain

(A.11) ζ (z) = F (−1)(z) = ζ+ϑ + c+ϑ z − Lϑ+ 1 + Aϑ+ z − Lϑ+

[with the convention Im F (−1)(z) ≥ 0], where Lϑ+ is defined by ζ+ϑ = Lϑ+ + mfc(Lϑ+). Here, c+ϑ > 0 is a real constant, and Aϑ+ is an analytic function that is real-valued on the real line and that satisfies Aϑ+(0) = 0. Recalling that ζ (z) = z + mϑfc(z) and taking the limit η → 0 we obtain (A.1), for fixed ϑ. To achieve uniformity in ϑ, we use the (uniform) stability bound (A.10) and the (pointwise) positivity of |F (ζ+ϑ )|: we differentiate (A.9) with respect to ϑ and observe that ∂ϑ H (ζ, ϑ)|ζ =ζ+ϑ = 0, for all ϑ ∈ , since F (ζ+ϑ ) = 0. Thus by the implicit function theorem, ζ+ϑ is a C1 function of ϑ ∈ . Next, we observe that F (ζ ) is an analytic function of ζ , for ζ away from ϑIν . Thus, using once more (A.10), we can bound |F (ζ+ϑ )| ≥ c, for some c > 0, uniformly in ϑ ∈ . In fact, F (n)(ζ+ϑ ), n ∈ N are all continuous functions of ϑ ∈ , and we can bound them uniformly in ϑ for each n ∈ N. Repeating the same argument for ζ close to ζ−ϑ , we complete the proof of (A.1).
Statement (2) follows from (A.10) for z close to the edges. For z away from the
edges, Assumption 2.3 assures that the curve γ stays away from the real line for
all ϑ ∈ as is readily checked. This implies the stability bound for that region.
For the proofs of the remaining statements, we refer to the Appendix
of [39].

Next, we prove Lemma 3.6.

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2417

PROOF OF LEMMA 3.6. It follows from Assumption 2.3 that on for all N sufficiently large,

(A.12)

1N

1

inf
x∈Iν

N

i=1

(ϑ vi

−

x)2

≥

1

+

/2,

for all ϑ ∈ = [0, 1+ /10]. The analogous statements of Lemma A.1, holding

on for N sufficiently large, follow in the same way as in the proof of that lemma.

To get uniformity in N , it suffices to check that the analogous expression to (A.10)

holds uniformly in N , for N sufficiency large: by (A.12) there are two real solu-

tions

ζ±ϑ

to

H (ζ )

:=

1 N

N i=1

1 (ϑvi −ζ )2

=

1

that

both

lie

outside

of

the

interval

ϑIν .

Thus (3.3) and (3.4) imply that

(A.13)

inf
ϑ∈

dist

ζ±ϑ , ϑ Iν

≥ g/2,

on for all N sufficiently large. Then we can bound

2N

1

F

(ζ )

=

− N

i=1

(ϑ vi

−

ζ )3 ,

evaluated at ζ±ϑ , uniformly below in ϑ and N , for N sufficiently large, implying the uniformity in N of the constants in statements (1)–(4).
Next we prove (3.22). For simplicity we drop ϑ from the notation and work on . As above, set ζ = z + mfc(z) and ζ = z + mfc(z). From the definitions of F , F and equations (3.5), (3.6), we have F (ζ ) = F (ζ ) = z, for all z ∈ D . Using the
stability bound (A.13) and equation (3.3) in the definition of , we get, assuming that |ζ − ζ | 1,

(A.14)

F (ζ ) + O N −α0

(ζ − ζ ) + F

(ζ ) (ζ

−

ζ )2

2

= o(1)(ζ − ζ )2 + O N −α0 ,

uniformly in ϑ ∈

,

for

all

z

∈

D

.

From

Lemma

A.1,

we

get

F

(ζ )

∼

√ κ

+

η

and F (ζ ) ≤ C, for all z ∈ D . We abbreviate := |ζ − ζ | in the following.

We first consider z = E + iη ∈ D , such that κE + η > N −ε, for some small

ε > 0 (with ε < α0). Here κE is defined in (A.2). For such z we obtain from (A.14) that ≤ CN ε( 2 + N −α0). Thus either ≤ C0N εN −α0 or C0N −ε ≤ , for some constant C0. We now show that | | ≤ C0N εN −α0 , for all z ∈ D such that κE + η ≥ N −ε. For z ∈ D with η = 2, we have

1N ζ (z) − ζ (z) =

ζ (z) − ζ (z)

+ O N −α0 ,

N i=1 (ϑvi − ζ (z))(ϑvi − ζ (z))

where we O(N −α0 ),

use that

(3.3). Since η = 2 is, (z) ≤ CN −α0 ,

and Im ζ , Im ζ ≥ η, for η = 2. To extend

we the

obtain

≤

1 4

conclusion to all

+ η,

2418

LEE, SCHNELLI, STETLER AND YAU

we use the Lipschitz continuity of ζ (z), respectively, ζ (z). Differentiating z = F (ζ ), with respect to z we obtain ∂zζ = (F (ζ ))−1. Thus using property (2) of
Lemma A.1, we infer that the Lipschitz constant of ζ (z) is, for z ∈ D satisfying κE + η > N −ε, bounded above by N ε/2. The same conclusion also holds for ζ (z).
Bootstrapping, we obtain

(A.15)

ζ (z) − ζ (z) ≤ CN εN −α0,

on for N sufficiently large, for all z ∈ D satisfying κE + η > N −ε. In order to control ζ (z) − ζ (z) for z = E + iη ∈ D with κE + η ≤ N −ε, ε > 0,
we first derive the estimate |Lϑ± − Lϑ±| ≤ CN −α0 , for some c > 0, on . We recall that L±, respectively, L±, are obtained through the relations

1N

1

N i=1 (ϑvi − ζ±)2 = 1,

dν(v) (ϑv − ζ±)2 = 1.

Then a similar argument as given above shows that |ζ± − ζ±| ≤ CN −α0 and |L± − L±| ≤ CN −α0 on , N sufficiently large. We refer to Section 4.3 in [40] for details.
Second, following the arguments in the proof of Lemma A.1, we may write, for ζ and ζ in a neighborhood of ζ±,

(A.16)

ζ (z) − ζ± = c± z − L± 1 + O(z − L±) ,

ζ (z) − ζ± = c+ z − L± 1 + O(z − L±) .

We

therefore

get

|ζ

(z)

−

ζ (z)|

≤

√ C κE

+

η

+

CN

−α0 /2 .

Note

that

the

constants

can be chosen uniformly in ϑ ∈ . Choosing, for example, ε = α0/4, we get

from (A.15) and (A.16) the desired inequality (3.22).

We now move on to the construction of the potentials U and U . We first record the following corollary of Lemma A.1. Set Br (p) := {z ∈ C : |z − p| < r}. Recall the conventions in (2.8) and the definition of κE in (A.2).

COROLLARY A.2. Under the assumptions of Lemma 3.5 there are constants

c+ϑ , r+ > 0, such that for any E ∈ Br+(Lϑ+) ∩ R,

(A.17)

Im mϑfc(E) =

√ κE

c+ϑ + B+ϑ (−κE)

,

0,

E ≤ Lϑ+, E ≥ Lϑ+,

and

(A.18)

Re mϑfc(E) =

√C+ϑκ(E−κc+ϑE

), +

B+ϑ (κE

)

+ C+ϑ (κE),

E ≤ Lϑ+, E ≥ Lϑ+,

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2419

where B+ϑ , C+ϑ are analytic functions on Br+(0) that are real-valued on R and that satisfy B+ϑ (0) = 0, c+ϑ +B+ϑ > 0, respectively, C+ϑ < 0, on Br+(0)∩R. Moreover, for all z ∈ Br+(Lϑ+), the functions B+ϑ , C+ϑ , respectively, Im mϑfc, Re mϑfc, are continuous in ϑ ∈ .
Similar statements hold at the lower edge Lϑ−.

PROOF. Fix ϑ ∈ . As argued in the proof of Lemma A.1, the function F (ζ ) can locally be inverted around ζ±ϑ ; see (A.11) above. Thus for ζ in a neighborhood of ζ+ϑ , we may write
mϑfc(z) = F −1(z) − z = ζ+ϑ − z + c+ϑ z − Lϑ+ 1 + Aϑ+ z − Lϑ+

= c+ϑ z − Lϑ+ 1 + B+ϑ z − Lϑ+ + C+ϑ z − Lϑ+ ,
for z in Br (Lϑ+), for some r > 0, where B+ϑ and C+ϑ are analytic in a neighborhood of zero and real-valued on the real line, since Im F −1(E) = 0, for E ∈ [Lϑ−, Lϑ+]c. Equations (A.17) and (A.18) follow. From the proof of Lemma A.1, it is immediate that c+ϑ > 0. Thus c+ϑ + B+ϑ > 0 in a real neighborhood of zero. Since x − Lϑ+ − mϑfc(Lϑ+) < 0, for all x ∈ ϑIν , we must have C+ϑ < 0 in a real neighborhood of zero. Since F (z) is analytic on Br (Lϑ+), for all ϑ ∈ , and since ζ+ϑ is a C1 function of ϑ, the functions B+ϑ and C+ϑ are C1 in ϑ ∈ . Then it is clear from (A.10) that we can choose r > 0 uniformly in ϑ ∈ . The same arguments apply for ζ close to ζ−ϑ .

The analogous result to Corollary A.2 is stated next. Recall the notation κE := min{|E − Lϑ−|, |E − Lϑ+|}.

COROLLARY A.3. Under the assumptions of Lemma 3.6 the following holds

on , for N sufficiently large. There are constants c+ϑ , r+, with r+ ≥ r+ > 0, such

that for any E ∈ Br+(Lϑ+) ∩ R,

(A.19)

Im mϑfc(E) =

√ κE

c+ϑ + B+ϑ (−κE)

,

0,

E ≤ Lϑ+, E ≥ Lϑ+,

and

(A.20)

Re mϑfc(E) =

√C+ϑκ(E−κc+ϑE

), +

B+ϑ (κE

)

+ C+ϑ (κE),

E ≤ Lϑ+, E ≥ Lϑ+,

where B+ϑ , C+ϑ are analytic functions on Br+(0) that are real-valued on R and that satisfy B+ϑ (0) = 0 and c+ϑ + B+ϑ > 0, respectively, C+ϑ < 0, on Br+(0) ∩ R. Moreover, the constant r+ can be chosen independent of ϑ ∈ and N , for N
sufficiently large.

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LEE, SCHNELLI, STETLER AND YAU

Further, the functions B+ϑ , C+ϑ , respectively, Im mϑfc, Re mϑfc, are continuous functions in ϑ ∈ δ, for all z ∈ Br+(Lϑ+). There is c > 0, such that

(A.21) B+ϑ (z) − B+ϑ (z) ≤ N −cα0/2,

C+ϑ (z) − C+ϑ (z) ≤ N −cα0/2,

for all z ∈ Br+(Lϑ+) and all ϑ ∈ , on for N sufficiently large. Similar statements hold at the lower edge Lϑ−.

PROOF. Corollary A.3 is proven in the same way as Corollary A.2. The only
things to be checked are that r± > 0 can be chosen uniformly in N , N sufficiently large, and the bounds in (A.21). The former statement is an immediate
consequence of the stability bound (A.13). The latter follows from z = F (ζ ) = F (ζ ), with ζ = z + mϑfc(z) and ζ = z + mϑfc(z). Then using (3.3), the stability bound (A.13) and the uniform lower bound on F (ζ±ϑ ), it is straightforward to derive estimate (A.21) from (3.22).

Next we prove Lemma 4.2. Recall from (4.16) that we chose ϑ ≡ ϑ(t) := e−(t −t0 )/2 .

PROOF OF LEMMA 4.2. For c > 0 and a measure ω on R, we define
suppc ω := supp ω + [−c, c]. Recall the constants r± > 0 of Corollary A.3. Set s := min{r−, r+}/2.
We specify the potentials U and U through their spatial derivatives U and U .
For t ≥ 0, we set

U (t, x) + x := −2− ρfc(t, y) dy, R y−x

U (t, x) + x := −2− ρfc(t, y) dy, R y−x

for x ∈ supp ρfc(t), respectively, x ∈ supp ρfc(t). For x ∈ R satisfying |x − L±(t)| ≤ s, where L±(t) denote the endpoints of the
support of the measure ρfc(t), we set

(A.22)

U (t, x) + x := −2C±ϑ (k±), U (t, x) + x := −2C±ϑ (k±),

k± ≡ x − L±(t), k± ≡ x − L±(t),

where C±ϑ are the functions appearing in Corollary A.3 with ϑ ≡ ϑ(t), and C±ϑ are the functions appearing in Corollary A.2 with ϑ ≡ ϑ(t). From Lemma A.1,
Corollaries A.2 and A.3, we conclude that U (t, x), respectively, U (t, x) are well
defined for x ∈ supps ρfc(t), t ≥ 0, where s = min{r−, r+}/2. For x ∈/ supps ρfc(t), we define U as a C3 extension in x such that:
(1) U (n)(t, x), ∂t U (n)(t, x), n ∈ [[1, 3]], are continuous in t ; (2) for all t ≥ 0 and
for all x ∈/ supps ρfc(t), |U (t, x) + x| > |2 Re mfc(t, x)| and U (t, x) ≥ −CU , for some constant CU ≥ 0; (3) U (t, x) + x ∼ x for all t ≥ 0, as |x| → ∞. Similarly, we define U (t, x) as C3 extensions such that: (1) U (n)(t, x), ∂t U (n)(t, x),

BULK UNIVERSALITY FOR DEFORMED WIGNER MATRICES

2421

n ∈ [[1, 3]], are continuous in t ; (2) there is c > 0 such that supt≥0 |U (n)(t, x) − U (n)(t, x)| ≤ N −cα0/2, n ∈ [[1, 3]], for N sufficiently large on .

We next show that the potential U (t, x) is a C3 function in x. For simplicity, we

often drop the t-dependence from the notation. Let ζ = z + mfc(z), and recall from the proof of Lemma A.1 that ζ (z) satisfies ζ (z) = F (−1)(z), where F (ζ ) = ζ −

dν(v) (ϑ v−ζ

)

.

Thus,

to

prove

regularity

of

U

(t, x)

in

x

in

the

support

of

the

measure

ρfc(t), it suffices to show that F (ζ ) = 0 on the curve γ ∩ C+ where Im F = 0.

Recall that on γ we have

(A.23)

H˜ (ζ ) :=

dν(v) |ϑv − ζ |2

=

1,

where ϑ ≡ ϑ(t). On the other hand, we have

Re F (ζ ) = 1 −

(ϑv − Re ζ )2 − (Im ζ )2

|ϑv − ζ |4

dν(v).

Thus, on the curve γ ,

Re F (ζ ) =

dν(v) |ϑv − ζ |2

−

(ϑv − Re ζ )2 − (Im ζ )2

|ϑv − ζ |4

dν(v)

=

2(Im ζ )2 |ϑv − ζ |4 dν(v).

From (2.10) we get

(A.24)

dν(v) |ϑv − ζ |4

≥

dν(v) |ϑv − ζ |2

2
= 1,

on γ . Since F = 0 on γ , the inverse function theorem implies that the real part
of mfc(t, x) is a smooth function in the interior of supp ρfc(t), whose derivatives are continuous in t. For x ∈ Bs(Lϑ±), we already showed in Lemma A.2 that C±ϑ (x) is a smooth function, whose derivatives are continuous in t. Thus we have shown that U (t, x) is smooth in supps ρfc(t). Outside supps ρfc(t), U (t, x) is manifestly C3 by definition: it is a C3 extension of the functions C±(t). Thus R x → U (t, x), ∂t U (t, x) are C3 functions for all t ≥ 0.
Clearly, we can bound the derivatives U (n)(t, x), ∂t U (n)(t, x), n ∈ [[1, 3]], uniformly on compact sets. It is also immediate that U (n)(t, x) are continuous functions in t ≥ 0. Thus we can bound U (n) uniformly in t and uniformly in x on
compact sets, for n ∈ [[1, 3]]. For x ∈ supps ρfc(t), we have U (t, x) ≥ −C, for some C ≥ 0. For x ∈/ supps ρfc(t), a similar bound holds true by construction. Thus U (t, x) satisfies (4.4) uniformly in t ≥ 0. Further, since U (t, x) + x ∼ x, as
|x| → ∞, (4.5) also holds uniformly in t ≥ 0.
On , we can extend the reasoning above to U (t, x), ∂t U (t, x), for N suffi-
ciently large. For example, the arguments in (A.23)–(A.24) can be extended to the

2422

LEE, SCHNELLI, STETLER AND YAU

finite N case by using (3.3) and Lemma 3.6. Let again s ≡ min{r−, r+}/2. Then for x ∈ supps ρfc(t) we have by Lemma 3.6 that |mfc(t, x + iη) − mfc(t, x + iη)| ≤ N −cα0 , for some c > 0, on for all η ≥ 0 and all t ≥ 0. Together with (A.21) we can conclude that |U (t, x) − U (t, x)| ≤ N −cα0/2 on , for x ∈ supps ρfc(t). We also have |∂xmfc(t, x +iη)−∂xmfc(t, x +iη)| ≤ CN −cα0 , for x satisfying min{|x − L+|, |x − L−|} ≥ s, as can be checked as in the proof of Lemma 3.6. Hence, combining this last statement with the regularity of C±ϑ claimed in Lemma A.3, we have |U (t, x) − U (t, x)| ≤ N −cα0 , for x ∈ supps ρfc(t), t ≥ 0, on for N sufficiently large. This conclusion can be extended to arbitrary U (n). Similarly, one checks that U (n)(t, x), n ∈ [[1, 3]] are continuous functions of t ≥ 0. For x ∈/ supps ρfc(t), these properties follow directly from the definition of U above. Thus U (t, x) sat-
isfies (4.4) and (4.5) with uniform constants for all t ≥ 0 and N sufficiently large
on .
Finally, the potentials U (t) and U (t) are “regular” as follows from Lemmas 3.5
and 3.6.

Acknowledgments. We thank Paul Bourgade, László Erdo˝s and Antti Knowles for helpful comments. We are grateful to the Taida Institute for Mathematical Sciences and National Taiwan Universality for their hospitality during part of this research. We thank Thomas Spencer and the Institute for Advanced Study for their hospitality during the academic year 2013–2014.

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J. O. LEE DEPARTMENT OF MATHEMATICAL SCIENCES KAIST 291 DAEHAK-RO YUSEONG-GU DAEJEON 305-701 REPUBLIC OF KOREA E-MAIL: jioon.lee@kaist.edu

K. SCHNELLI INSTITUTE OF SCIENCE AND TECHNOLOGY AUSTRIA AM CAMPUS 1 3400 KLOSTERNEUBURG REPUBLIC OF AUSTRIA E-MAIL: kevin.schnelli@ist.ac.at

B. STETLER H.-T. YAU DEPARTMENT OF MATHEMATICS HARVARD UNIVERSITY ONE OXFORD STREET CAMBRIDGE, MASSACHUSETTS 02138 USA E-MAIL: bstetler@math.harvard.edu
htyau@math.harvard.edu