ICTS

Dyson introduced the Coulomb gas models via a Brownian matrix  process for the eigenvalues of a random matrix. Dyson showed that joint probability distributions (jpd) of the classical ensembles are equilibrium states of the Brownian motion model of N Coulombic particles interacting with each other. The positions of  the Brownian particles were identified as the eigenvalues of the random matrix problem. In the Dyson model the particles had full range or infinite-range interaction. Coulomb gas models were thus  shown to follow the classical random matrix-universality classes. In this talk I will discusse the finite-range Coulomb gas (FRCG) models where the range of eigenvalue interaction d is finite. It can be shown that as the range of interaction increases from (d=0) to  (d=N-1), there is a transition from Poisson to Wigner-Dyson classical random matrix statistics. FRCG models yield new universality classes for each d, and provide a theoretical framework for the study of banded random matrices (BRMs), and the dynamical systems (such as quantum kicked rotors (QKRs)) whose matrix representation can be written in the form of banded natrices.