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Now showing items 1-10 of 24

#### 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 ...

#### 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 ...

#### 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 ...

#### 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 ...

#### 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 ...

#### 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 ...

#### 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 ...

#### 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 ...

#### 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 ...

#### 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 ...