Browsing by Author "Erdos, Laszlo"
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Bulk universality for generalized Wigner matrices
Erdos, Laszlo; Yau, HorngTzer; Yin, Jun (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
Erdos, Laszlo; Péché, Sandrine; Ramírez, José A.; Schlein, Benjamin; Yau, HorngTzer (WileyBlackwell, 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 general βensembles with nonconvex potential
Bourgade, Paul; Erdos, Laszlo; Yau, HorngTzer (AIP Publishing, 201209)We prove the bulk universality of the βensembles with nonconvex 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, ... 
Derivation of the cubic nonlinear Schrödinger equation from quantum dynamics of manybody systems
Erdos, Laszlo; Schlein, Benjamin; Yau, HorngTzer (Springer Nature, 2006)We prove rigorously that the oneparticle density matrix of three dimensional interacting Bose systems with a shortscale repulsive pair interaction converges to the solution of the cubic nonlinear Schrödinger equation ... 
Derivation of the GrossPitaevskii equation for the dynamics of BoseEinstein condensate
Erdos, Laszlo; Schlein, Benjamin; Yau, HorngTzer (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 ... 
Feynman Graphs and Renormalization in Quantum Diffusion
Erdos, Laszlo; Salmhofer, Manfred; Yau, HorngTzer (World Scientific Publishing, 2008)We review our proof that in a scaling limit, the time evolution of a quantum particle in a static random environment leads to a diffusion equation. In particular, we discuss the role of Feynman graph expansions and of ... 
Fixed Energy Universality for Generalized Wigner Matrices
Bourgade, Paul; Erdos, Laszlo; Yau, HorngTzer; Yin, Jun (WileyBlackwell, 2015)We prove the WignerDysonMehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and Hermitian Wigner matrices. Previous results concerning the universality of random matrices either require ... 
GrossPitaevskii Equation as the Mean Field Limit of Weakly Coupled Bosons
Elgart, Alexander; Erdos, Laszlo; Schlein, Benjamin; Yau, HorngTzer (Springer Nature, 2005)We consider the dynamics of N boson systems interacting through a pair potential N−1Va(xi−xj) where Va(x)=a−3V(x/a). We denote the solution to the Nparticle Schrödinger equation by ΨN, t. Recall that the GrossPitaevskii ... 
Isotropic local laws for sample covariance and generalized Wigner matrices
Alex, Bloemendal; Erdos, Laszlo; Knowles, Antti; Yau, HorngTzer; Yin, Jun (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
Erdos, Laszlo; Yau, HorngTzer (WileyBlackwell, 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
Erdos, Laszlo; Schlein, Benjamin; Yau, HorngTzer (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 ... 
Nonlinear Hartree Equation as the Mean Field Limit of Weakly Coupled Fermions
Elgart, Alexander; Erdos, Laszlo; Schlein, Benjamin; Yau, HorngTzer (Elsevier BV, 2004)We consider a system of N weakly interacting fermions with a real analytic pair interaction. We prove that for a general class of initial data there exists a fixed time T such that the difference between the one particle ... 
On the Quantum Boltzmann Equation
Erdos, Laszlo; Salmhofer, Manfred; Yau, HorngTzer (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
Erdos, Laszlo; Salmhofer, Manfred; Yau, HorngTzer (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
Erdos, Laszlo; Salmhofer, Manfred; Yau, HorngTzer (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
Erdos, Laszlo; Salmhofer, Manfred; Yau, HorngTzer (Springer Nature, 2007)We consider random Schrödinger equations on {mathbb{R}d} for d≥ 3 with a homogeneous AndersonPoisson 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
Erdos, Laszio; Yau, HorngTzer; Yin, Jun (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 ErdősRényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues
Erdos, Laszlo; Knowles, Antti; Yau, HorngTzer; Yin, Jun (Springer Nature, 2012)We consider the ensemble of adjacency matrices of ErdősRényi 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 ... 
Towards the Quantum Brownian Motion
Erdos, Laszlo; Salmhofer, Manfred; Yau, HorngTzer (Springer Berlin Heidelberg, 2006)We consider random Schr\"odinger equations on $\bR^d$ or $\bZ^d$ for d≥3 with uncorrelated, identically distributed random potential. Denote by λ the coupling constant and ψt the solution with initial data ψ0. Suppose that ... 
Universality of general βensembles
Bourgade, Paul; Erdos, Laszlo; Yau, HorngTzer (Duke University Press, 2014)We prove the universality of the βensembles with convex analytic potentials and for any β>0; that is, we show that the spacing distributions of loggases at any inverse temperature β coincide with those of the Gaussian ...