Person:

Erdos, Laszlo

We prove the universality of the β-ensembles with convex analytic potentials and for any β>0; that is, we show that the spacing distributions of log-gases at any inverse temperature β coincide with those of the Gaussian β-ensembles.

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.

We give a nonrigorous derivation of the nonlinear Boltzmann equation from the Schrödinger evolution of interacting fermions. The argument is based mainly on the assumption that a quasifree initial state satisfies a property called restricted quasifreenessin the weak coupling limit at any later time. By definition, a state is called restricted quasifree if the four-point and the eight-point functions of the state factorize in the same manner as in a quasifree state.

We consider N×N Hermitian random matrices with independent identically distributed entries (Wigner matrices). We assume that the distribution of the entries have a Gaussian component with variance N−3/4+βN−3/4+β for some positive β>0β>0. We prove that the local eigenvalue statistics follows the universal Dyson sine kernel.

We consider random Schrödinger equations on ℤdZd for d ≥ 3 with identically distributed random potential. Denote by λ the coupling constant and ψt the solution with initial data ψ0. The space and time variables scale as x∼λ−2−κ/2,t∼λ−2−κx∼λ−2−κ/2,t∼λ−2−κ with 0 < κ < κ0(d). We prove that, in the limit λ → 0, the expectation of the Wigner distribution of ψt converges weakly to a solution of a heat equation in the space variable x for arbitrary L2 initial data. The diffusion coefficient is uniquely determined by the kinetic energy associated to the momentum υ.

This work is an extension to the lattice case of our previous result in the continuum [8,9]. Due to the non-convexity of the level surfaces of the dispersion relation, the estimates of several Feynman graphs are more involved.

We consider N × N Hermitian random matrices with independent identical distributed entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order 1/N. Under suitable assumptions on the distribution of the single matrix element, we prove that, away from the spectral edges, the density of eigenvalues concentrates around the Wigner semicircle law on energy scales η≫N−1(logN)8η≫N−1(log⁡N)8 . Up to the logarithmic factor, this is the smallest energy scale for which the semicircle law may be valid. We also prove that for all eigenvalues away from the spectral edges, the ℓ∞-norm of the corresponding eigenvectors is of order O(N−1/2), modulo logarithmic corrections. The upper bound O(N−1/2) implies that every eigenvector is completely delocalized, i.e., the maximum size of the components of the eigenvector is of the same order as their average size.

In the Appendix, we include a lemma by J. Bourgain which removes one of our assumptions on the distribution of the matrix elements.

We consider N × N Hermitian Wigner random matrices H where the probability density for each matrix element is given by the density ν(x) = e−U(x). We prove that the eigenvalue statistics in the bulk are given by the Dyson sine kernel provided that U ∈ C6( \input amssym $\Bbb R$) with at most polynomially growing derivatives and ν(x) ≥ Ce−C|x| for x large. The proof is based upon an approximate time reversal of the Dyson Brownian motion combined with the convergence of the eigenvalue density to the Wigner semicircle law on short scales. © 2010 Wiley Periodicals, Inc.

Consider a system of N bosons in three dimensions interacting via a repulsive short range pair potential N²V (N(xi − xj)), where x = (x1,..., xN) denotes the positions of the particles. Let HN denote the Hamiltonian of the system and let ψN,t be the solution to the Schrödinger equation. Suppose that the initial data ψN,0 satisfies the energy condition 〈ψN,0, H k NψN,0 〉 ≤ C k N k for k = 1, 2,.... We also assume that the k-particle density matrices of the initial state are asymptotically factorized as N → ∞. We prove that the k-particle density matrices of ψN,t are also asymptotically factorized and the one particle orbital wave function solves the Gross-Pitaevskii equation, a cubic non-linear Schrödinger equation with the coupling constant given by the scattering length of the potential V. We also prove the same conclusion if the energy condition holds only for k = 1 but the factorization of ψN,0 is assumed in a stronger sense.