Now showing items 1-17 of 17

• #### 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 ...
• #### 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 ...
• #### Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems ﻿

(Springer Nature, 2006)
We prove rigorously that the one-particle density matrix of three dimensional interacting Bose systems with a short-scale repulsive pair interaction converges to the solution of the cubic non-linear Schrödinger equation ...
• #### 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 ...
• #### 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 ...
• #### Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation ﻿

(Wiley-Blackwell, 2000)
We study the time evolution of a quantum particle in a Gaussian random environment. We show that in the weak coupling limit the Wigner distribution of the wave function converges to a solution of a linear Boltzmann equation ...
• #### Local Semicircle Law and Complete Delocalization for Wigner Random Matrices ﻿

(Springer Nature, 2008)
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 Quantum Boltzmann Equation ﻿

(Springer Nature, 2004)
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 ...
• #### Quantum Diffusion for the Anderson Model in the Scaling Limit ﻿

(Springer Nature, 2007)
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 ...
• #### Quantum diffusion of the random Schrödinger evolution in the scaling limit ﻿

(International Press of Boston, 2008)
We consider random Schrödinger equations on Rd for d ≽ 3 with a homogeneous Anderson–Poisson type random potential. Denote by λ the coupling constant and ψtψt the solution with initial data ψ0ψ0 . The space and time ...
• #### Quantum Diffusion of the Random Schrödinger Evolution in the Scaling Limit II. The Recollision Diagrams ﻿

(Springer Nature, 2007)
We consider random Schrödinger equations on {mathbb{R}d} for d≥ 3 with a homogeneous Anderson-Poisson type random potential. Denote by λ the coupling constant and ψ t the solution with initial data ψ0. The space and time ...
• #### 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 ...
• #### 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 ...
• #### 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 ...
• #### Universality of random matrices and local relaxation flow ﻿

(Springer Nature, 2010)
Consider the Dyson Brownian motion with parameter β, where β=1,2,4 corresponds to the eigenvalue flows for the eigenvalues of symmetric, hermitian and quaternion self-dual ensembles. For any β≥1, we prove that the relaxation ...
• #### 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 ...
• #### Wegner Estimate and Level Repulsion for Wigner Random Matrices ﻿

(Oxford University Press (OUP), 2009)
We consider N × N Hermitian random matrices with independent identically distributed entries (Wigner matrices). The matrices are normalized so that the average spacing between consecutive eigenvalues is of order 1/ N. Under ...