Browsing by Author "Erdos, Laszlo"
Now showing items 1-20 of 23
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Bulk universality for generalized Wigner matrices
Erdos, Laszlo; Yau, Horng-Tzer; 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, Horng-Tzer (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 general β-ensembles with non-convex potential
Bourgade, Paul; Erdos, Laszlo; Yau, Horng-Tzer (AIP Publishing, 2012-09)We prove the bulk universality of the β-ensembles with non-convex 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 non-linear Schrödinger equation from quantum dynamics of many-body systems
Erdos, Laszlo; Schlein, Benjamin; Yau, Horng-Tzer (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
Erdos, Laszlo; Schlein, Benjamin; Yau, Horng-Tzer (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 ... -
Fixed Energy Universality for Generalized Wigner Matrices
Bourgade, Paul; Erdos, Laszlo; Yau, Horng-Tzer; Yin, Jun (Wiley-Blackwell, 2015)We prove the Wigner-Dyson-Mehta 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 ... -
Gross-Pitaevskii Equation as the Mean Field Limit of Weakly Coupled Bosons
Elgart, Alexander; Erdos, Laszlo; Schlein, Benjamin; Yau, Horng-Tzer (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 N-particle Schrödinger equation by ΨN, t. Recall that the Gross-Pitaevskii ... -
Isotropic local laws for sample covariance and generalized Wigner matrices
Alex, Bloemendal; Erdos, Laszlo; Knowles, Antti; Yau, Horng-Tzer; 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, Horng-Tzer (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
Erdos, Laszlo; Schlein, Benjamin; Yau, Horng-Tzer (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, Horng-Tzer (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, Horng-Tzer (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, Horng-Tzer (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, Horng-Tzer (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, Horng-Tzer (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
Erdos, Laszio; Yau, Horng-Tzer; 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 Erdős-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues
Erdos, Laszlo; Knowles, Antti; Yau, Horng-Tzer; Yin, Jun (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 ... -
Towards the Quantum Brownian Motion
Erdos, Laszlo; Salmhofer, Manfred; Yau, Horng-Tzer (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 β-ensembles
Bourgade, Paul; Erdos, Laszlo; Yau, Horng-Tzer (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 local spectral statistics of random matrices
Erdos, Laszlo; Yau, Horng-Tzer (American Mathematical Society (AMS), 2012-01-30)The Wigner-Gaudin-Mehta-Dyson conjecture asserts that the local eigenvalue statistics of large random matrices exhibit universal behavior depending only on the symmetry class of the matrix ensemble. For invariant matrix ...