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Bulk universality for generalized Wigner matrices
(Springer Science + Business Media, 2011)
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 ...
Rigidity of Eigenvalues of Generalized Wigner Matrices
(Elsevier BV, 2012)
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 ...
Universality of Sine-Kernel for Wigner Matrices with a Small Gaussian Perturbation
(Institute of Mathematical Statistics, 2010)
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 ...
Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate
(Annals of Mathematics, Princeton U, 2010)
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 ...
Bulk universality for Wigner matrices
(Wiley-Blackwell, 2010)
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 ...
Universality of general β-ensembles
(Duke University Press, 2014)
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 ...
Bulk universality of general β-ensembles with non-convex potential
(AIP Publishing, 2012-09)
We prove the bulk universality of the β-ensembles with non-convex regular analytic potentials for any β > 0. This removes the convexity assumption appeared in the earlier work [P. Bourgade, L. Erdös, and H.-T. Yau, ...
Spectral Statistics of Erdős-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues
(Springer Nature, 2012)
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 ...
Isotropic local laws for sample covariance and generalized Wigner matrices
(Institute of Mathematical Statistics, 2014)
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 ...
Universality of local spectral statistics of random matrices
(American Mathematical Society (AMS), 2012-01-30)
The Wigner-Gaudin-Mehta-Dyson conjecture asserts that the local eigenvalue statistics of large random matrices exhibit universal behavior depending only on the symmetry class of the matrix ensemble. For invariant matrix ...