Now showing items 1-20 of 42

• Asymptotic dynamics of nonlinear Schrödinger equations: Resonance-dominated and dispersion-dominated solutions ﻿

(Wiley-Blackwell, 2001)
We consider a linear Schrödinger equation with a nonlinear perturbation in ℝ3. Assume that the linear Hamiltonian has exactly two bound states and its eigen-values satisfy some resonance condition. We prove that if the ...
• Bulk diffusivity of lattice gases close to criticality ﻿

(Springer Nature, 1995)
We consider lattice gases where particles jump at random times constrained by hard-core exclusion (simple exclusion process with speed change). The conventional theory of critical slowing down predicts that close to a ...
• Bulk universality for deformed Wigner matrices ﻿

(Institute of Mathematical Statistics, 2016)
We consider N×N random matrices of the form H=W+V where W is a real symmetric or complex Hermitian Wigner matrix and V is a random or deterministic, real, diagonal matrix whose entries are independent of W. We assume ...
• 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 ...
• Bulk universality of sparse random matrices ﻿

(AIP Publishing, 2015)
We consider the adjacency matrix of the ensemble of Erdős-Rényi random graphs which consists of graphs on N vertices in which each edge occurs independently with probability p. We prove that in the regime pN ≫ 1, these ...
• Convergence to equilibrium of conservative particle systems on ℤ\bmd ﻿

(Institute of Mathematical Statistics, 2003)
We consider the Ginzburg--Landau process on the lattice ℤdZd whose potential is a bounded perturbation of the Gaussian potential. We prove that the decay rate to equilibrium in the variance sense is t−d/2t−d/2 up to ...
• 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 ...
• Diffusive limit of lattice gas with mixing conditions ﻿

(International Press of Boston, 1997)
We prove, under certain mixing conditions, that the hydrodynamical limit of a stochastic lattice gas on the cubic lattice Z d is governed by a nonlinear diffusion equation. Following [VI], we characterize the diffusion ...
• 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 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 ...
• The local relaxation flow approach to universality of the local statistics for random matrices ﻿

(Institute of Mathematical Statistics, 2012)
We present a generalization of the method of the local relaxation flow to establish the universality of local spectral statistics of a broad class of large random matrices. We show that the local distribution 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 ...
• Logarithmic Sobolev inequality for lattice gases with mixing conditions ﻿

(Springer Nature, 1996)
Let μgcΛL,λμΛL,λgc denote the grand canonical Gibbs measure of a lattice gas in a cube of sizeL with the chemical potential γ and a fixed boundary condition. Let μcΛL,nμΛL,nc be the corresponding canonical measure defined ...
• Logarithmic Sobolev inequality for some models of random walks ﻿

(Institute of Mathematical Statistics, 1998)
We determine the logarithmic Sobolev constant for the Bernoulli- Laplace model and the time to stationarity for the symmetric simple exclusion model up to the leading order. Our method for proving the logarithmic Sobolev ...
• (logt)2/3 law of the two dimensional asymmetric simple exclusion process ﻿

(Annals of Mathematics, Princeton U, 2004)
We prove that the diffusion coefficient for the two dimensional asymmetric simple exclusion process diverges as (logt)2/3 to the leading order. The method applies to nearest and non-nearest neighbor asymmetric simple ...
• Lower Bound on the Blow-up Rate of the Axisymmetric Navier-Stokes Equations ﻿

(Oxford University Press (OUP), 2010)
Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in R 3 with non-trivial swirl. Such solutions are not known to be globally defined, but it is shown in [11, 1] that they could only blow ...