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 ... 
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 ... 
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 ... 
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 ... 
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 ... 
Universality of random matrices and local relaxation flow
Erdos, Laszlo; Schlein, Benjamin; Yau, HorngTzer (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 selfdual ensembles. For any β≥1, we prove that the relaxation ... 
Universality of SineKernel for Wigner Matrices with a Small Gaussian Perturbation
Erdos, Laszlo; Ramirez, Jose; Schlein, Benjamin; Yau, HorngTzer (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
Erdos, L; Schlein, B.; Yau, HorngTzer (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 ...