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Yau, Horng-Tzer

Consider (N) bosons in a finite box (\Lambda= [0,L]^3\subset \mathbf R^3) interacting via a two-body smooth repulsive short range potential. We construct a variational state which gives the following upper bound on the ground state energy per particle [\overline\lim_{\rho\to0} \overline\lim_{L \to \infty, N/L^3 \to \rho} \left(\frac{e_0(\rho)- 4 \pi a \rho}{(4 \pi a)^{5/2}(\rho)^{3/2}}\right)\leq \frac{16}{15\pi^2}, ] where (a) is the scattering length of the potential. Previously, an upper bound of the form (C 16/15\pi^2) for some constant (C > 1) was obtained in. Our result proves the upper bound of the the prediction by Lee-Yang and Lee-Huang-Yang.

We present a generalization of the method of the local relaxation flow to establish the universality of local spectral statistics of a broad class of large random matrices. We show that the local distribution of the eigenvalues coincides with the local statistics of the corresponding Gaussian ensemble provided the distribution of the individual matrix element is smooth and the eigenvalues {(x_{j})}({j=1}^{N}) are close to their classical location {(\gamma)({j})}({j=1}^{N}) determined by the limiting density of eigenvalues. Under the scaling where the typical distance between neighboring eigenvalues is of order 1/(N), the necessary apriori estimate on the location of eigenvalues requires only to know that (\mathbb{E}) |(x{j}) (-) (\gamma)(_{j})|(^{2}) (\leq) (N)(^{-1-\epsilon}) on average. This information can be obtained by well established methods for various matrix ensembles. We demonstrate the method by proving local spectral universality for Wishart matrices.

We introduce a class of M×MM×M sample covariance matrices Q which subsumes and generalizes several previous models. The associated population covariance matrix Σ=[E] Σ=EQ is assumed to differ from the identity by a matrix of bounded rank. All quantities except the rank of Σ−IMΣ−IM may depend on MM in an arbitrary fashion. We investigate the principal components, i.e. the top eigenvalues and eigenvectors, of Q . We derive precise large deviation estimates on the generalized components [] of the outlier and non-outlier eigenvectors [] . Our results also hold near the so-called BBP transition, where outliers are created or annihilated, and for degenerate or near-degenerate outliers. We believe the obtained rates of convergence to be optimal. In addition, we derive the asymptotic distribution of the generalized components of the non-outlier eigenvectors. A novel observation arising from our results is that, unlike the eigenvalues, the eigenvectors of the principal components contain information about the subcritical spikes of ΣΣ . The proofs use several results on the eigenvalues and eigenvectors of the uncorrelated matrix Q , satisfying [E] =IMEQ=IM , as input: the isotropic local Marchenko–Pastur law established in Bloemendal et al. (Electron J Probab 19:1–53, 2014), level repulsion, and quantum unique ergodicity of the eigenvectors. The latter is a special case of a new universality result for the joint eigenvalue–eigenvector distribution.

Consider (N × N) Hermitian or symmetric random matrices H where the distribution of the (i, j) matrix element is given by a probability measure (\nu_{ij}) with a subexponential decay. Let (\sigma_{ij}^2) be the variance for the probability measure (\nu_{ij}) with the normalization property that (\sum_i\sigma_{ij}^2 = 1) for all j. Under essentially the only condition that (c\leq N\sigma_{ij}^2 \leq c^{−1}) for some constant (c > 0), we prove that, in the limit (N \rightarrow \infty), the eigenvalue spacing statistics of H in the bulk of the spectrum coincide with those of the Gaussian unitary or orthogonal ensemble (GUE or GOE). We also show that for band matrices with bandwidth M the local semicircle law holds to the energy scale (M^{−1}).

Consider (N\times N) hermitian or symmetric random matrices (H) with independent entries, where the distribution of the ((i,j)) matrix element is given by the probability measure (\nu_{ij}) with zero expectation and with variance (\sigma_{ij}^2). We assume that the variances satisfy the normalization condition (\sum_{i} \sigma^2_{ij} = 1) for all (j) and that there is a positive constant (c) such that (c\le N \sigma_{ij}^2 \le c^{-1}). We further assume that the probability distributions (\nu_{ij}) have a uniform subexponential decay. We prove that the Stieltjes transform of the empirical eigenvalue distribution of (H) is given by the Wigner semicircle law uniformly up to the edges of the spectrum with an error of order ( (N \eta)^{-1}) where (\eta) is the imaginary part of the spectral parameter in the Stieltjes transform. There are three corollaries to this strong local semicircle law: (1) Rigidity of eigenvalues: If (\gamma_j =\gamma_{j,N}) denotes the classical location of the (j)-th eigenvalue under the semicircle law ordered in increasing order, then the (j)-th eigenvalue (\lambda_j) is close to (\gamma_j) in the sense that for some positive constants (C, c) (\mathbb P \Big (\exists , j : ; |\lambda_j-\gamma_j| \ge (\log N)^{C\ log\ log\ N} \Big [ \min \big (, j, N-j+1 , \big) \Big ]^{-1/3} N^{-2/3} \Big) \le C\exp{\big[-c(\log N)^{c\ log\ log\ N} \big]}) for (N) large enough. (2) The proof of the Dyson's conjecture which states that the time scale of the Dyson Brownian motion to reach local equilibrium is of order (N^{-1}). (3) The edge universality holds in the sense that the probability distributions of the largest (and the smallest) eigenvalues of two generalized Wigner ensembles are the same in the large (N) limit provided that the second moments of the two ensembles are identical.

We consider sample covariance matrices of the form X ∗X, where X is an M × N matrix with independent random entries. We prove the isotropic local MarchenkoPastur law, i.e. we prove that the resolvent (X ∗X − z) −1 converges to a multiple of the identity in the sense of quadratic forms. More precisely, we establish sharp high-probability bounds on the quantity hv,(X ∗X − z) −1wi − hv, wim(z), where m is the Stieltjes transform of the Marchenko-Pastur law and v, w ∈ C N . We require the logarithms of the dimensions M and N to be comparable. Our result holds down to scales Im z > N −1+ε and throughout the entire spectrum away from 0. We also prove analogous results for generalized Wigner matrices.

In the first part of this article (Bourgade et al. arXiv:1206.1449, 2012), we proved a local version of the circular law up to the finest scale N−1/2+εN−1/2+ε for non-Hermitian random matrices at any point z∈ℂz∈C with ||z|−1|>c||z|−1|>c for any c>0c>0 independent of the size of the matrix. Under the main assumption that the first three moments of the matrix elements match those of a standard Gaussian random variable after proper rescaling, we extend this result to include the edge case |z|−1=o(1)|z|−1=o(1). Without the vanishing third moment assumption, we prove that the circular law is valid near the spectral edge |z|−1=o(1)|z|−1=o(1) up to scale N−1/4+εN−1/4+ε.

The circular law asserts that the spectral measure of eigenvalues of rescaled random matrices without symmetry assumption converges to the uniform measure on the unit disk. We prove a local version of this law at any point zz away from the unit circle. More precisely, if ||z|−1|≥τ||z|−1|≥τ for arbitrarily small τ>0τ>0 , the circular law is valid around zz up to scale N−1/2+εN−1/2+ε for any ε>0ε>0 under the assumption that the distributions of the matrix entries satisfy a uniform subexponential decay condition.

We consider the ensemble of adjacency matrices of Erdős-Rényi random graphs, i.e. graphs on N vertices where every edge is chosen independently and with probability p ≡ p(N). We rescale the matrix so that its bulk eigenvalues are of order one. Under the assumption pN≫N2/3pN≫N2/3 , we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum. More precisely, we prove (1) that the eigenvalue spacing of the Erdős-Rényi graph in the bulk of the spectrum has the same distribution as that of the Gaussian orthogonal ensemble; and (2) that the second largest eigenvalue of the Erdős-Rényi graph has the same distribution as the largest eigenvalue of the Gaussian orthogonal ensemble. As an application of our method, we prove the bulk universality of generalized Wigner matrices under the assumption that the matrix entries have at least 4 + ε moments.

We prove the Wigner-Dyson-Mehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and Hermitian Wigner matrices. Previous results concerning the universality of random matrices either require an averaging in the energy parameter or they hold only for Hermitian matrices if the energy parameter is fixed. We develop a homogenization theory of the Dyson Brownian motion and show that microscopic universality follows from mesoscopic statistics.