Now showing items 1-7 of 7

• #### 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 ...
• #### 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 ...
• #### Local circular law for random matrices ﻿

(Springer Nature, 2013)
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 ...
• #### The local circular law II: the edge case ﻿

(Springer Nature, 2013)
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 ...
• #### A Lower Bound on the Ground State Energy of Dilute Bose Gas ﻿

(American Institute of Physics, 2010)
Consider an N-Boson system interacting via a two-body repulsive short-range potential ${V}$ in a three dimensional box ${\Lambda}$ of side length ${L}$. We take the limit ${N}$, ${L}$ ${\rightarrow}$ ${\infty}$ ...
• #### 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 ...