# Finite-Range Coulomb Gas Models II: Applications to Quantum Kicked Rotors and Banded Random Matrices

## TL;DR

This paper develops Monte Carlo methods to study finite-range Coulomb gas models, revealing a transition from Poisson to Wigner-Dyson statistics, and applies these models to banded random matrices and quantum kicked rotors.

## Contribution

It introduces a Monte Carlo approach for arbitrary interaction range in FRCG models and connects these models to banded random matrices and quantum kicked rotors.

## Key findings

- Transition from Poisson to Wigner-Dyson statistics as range increases
- Monte Carlo simulations effectively study FRCG models for any range
- FRCG models describe spectral properties of banded matrices and quantum rotors

## Abstract

In paper I of this two-stage exposition, we introduced finite-range Coulomb gas (FRCG) models, and developed an integral-equation framework for their study. We obtained exact analytical results for $d = 0,1,2 $, where d denotes the range of eigenvalue interaction. We found that the integral-equation framework was not analytically tractable for higher values of $d$. In this paper II, we develop a Monte Carlo (MC) technique to study FRCG models. Our MC simulations provide a solution of FRCG models for arbitrary $d$. We show that, as d increases, there is a transition from Poisson to Wigner-Dyson classical random matrix statistics. Thus FRCG models provide a novel route for transition from Poisson to Wigner-Dyson statistics. The analytical formulation obtained in paper I, and MC techniques developed in this paper II, are used to study banded random matrices (BRM) and quantum kicked rotors (QKR). We demonstrate that, for a BRM of bandwidth $b$ and a QKR of chaos parameter $\alpha$, the appropriate FRCG model has range $d=b^2/N=\alpha^2/N$, for $N \rightarrow \infty $. Here, N is the dimensionality of the matrix in BRM, and the evolution operator matrix in QKR.