Browsing by Author "Yau, HorngTzer"
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Asymptotic dynamics of nonlinear Schrödinger equations: Resonancedominated and dispersiondominated solutions
Tsai, TaiPeng; Yau, HorngTzer (WileyBlackwell, 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 eigenvalues satisfy some resonance condition. We prove that if the ... 
Bulk diffusivity of lattice gases close to criticality
Spohn, Herbert; Yau, HorngTzer (Springer Nature, 1995)We consider lattice gases where particles jump at random times constrained by hardcore 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
Lee, Ji Oon; Schnelli, Kevin; Stetler, Benjamin; Yau, HorngTzer (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
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, ... 
Bulk universality of sparse random matrices
Huang, Jiaoyang; Landon, Benjamin; Yau, HorngTzer (AIP Publishing, 2015)We consider the adjacency matrix of the ensemble of ErdősRényi 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
Landim, C.; Yau, HorngTzer (Institute of Mathematical Statistics, 2003)We consider the GinzburgLandau 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 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 ... 
Diffusive limit of lattice gas with mixing conditions
Varadhan, S. R. S.; Yau, HorngTzer (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 ... 
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 circular law for random matrices
Bourgade, Paul; Yau, HorngTzer; Yin, Jun (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
Bourgade, Paul; Yau, HorngTzer; Yin, Jun (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 nonHermitian random matrices at any point z∈ℂz∈C with ... 
The local relaxation flow approach to universality of the local statistics for random matrices
Schlein, Benjamin; Yau, HorngTzer; Yin, Jun (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
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 ...