Asymptotic Level Spacing of the Laguerre Ensemble: A Coulomb Fluid Approach

TL;DR

This paper derives the asymptotic level spacing distribution for the Laguerre Ensemble using Coulomb Fluid methods, reproducing known results and extending them to include finite temperature effects without relying on Painlevé equations.

Contribution

It introduces a Coulomb Fluid approach to asymptotic level spacing in the Laguerre Ensemble, recovering known results and deriving new finite temperature corrections.

Findings

Reproduces Tracy-Widom leading terms for $eta=2$

Extends results to $eta eq 2$ with finite temperature corrections

Provides an alternative to Painlevé-based derivations

Abstract

We determine the asymptotic level spacing distribution for the Laguerre Ensemble in a single scaled interval, $(0,s)$, containing no levels, $E_{\bt}(0,s)$, via Dyson's Coulomb Fluid approach. For the $\alpha=0$ Unitary-Laguerre Ensemble, we recover the exact spacing distribution found by both Edelman and Forrester, while for $\alpha\neq 0$, the leading terms of $E_{2}(0,s)$, found by Tracy and Widom, are reproduced without the use of the Bessel kernel and the associated Painlev\'e transcendent. In the same approximation, the next leading term, due to a ``finite temperature'' perturbation $(\bt\neq 2)$, is found.