Large deviation principle in random matrices

Abstract

Many important properties of physical systems can be represented mathematically as matrix problems. For example, the thermal conductivity of a lattice can be computed from the dynamical matrix of the particle-particle interactions within the lattice. Consider N by N complex Hermitian or real symmetric random matrices H whose upper right entries are i.i.d. random variables. It is well-known that, under suitable conditions such as subexponential decay, the local semi-circle law for eigenvalues and the delocalization of eigenvectors hold with high probability. In this paper, we study the relation between large deviation estimates and the probability with which the results for the random matrices hold. A detailed proof for the improved large deviation estimates for random matrices is also given.