# Finite-Range Coulomb Gas Models I: Some Analytical Results

## TL;DR

This paper introduces finite-range Coulomb gas models that interpolate between Poisson and classical random matrix statistics, revealing new universality classes and providing insights into banded matrices and related dynamical systems.

## Contribution

It generalizes Dyson's models by incorporating finite-range interactions, establishing a framework for new universality classes in random matrix theory.

## Key findings

- Transition from Poisson to classical RMT statistics with increasing interaction range
- Introduction of new universality classes in random matrix ensembles
- Application to banded random matrices and dynamical systems

## Abstract

Dyson has shown an equivalence between infinite-range Coulomb gas models and classical random matrix ensembles for the study of eigenvalue statistics. In this paper, we introduce finite-range Coulomb gas (FRCG) models as a generalization of the Dyson models with a finite range of eigenvalue interactions. As the range of interaction increases, there is a transition from Poisson statistics to classical random matrix statistics. These models yield new universality classes of random matrix ensembles. They also provide a theoretical framework to study banded random matrices, and dynamical systems whose matrix representation can be written in the form of banded matrices.