Universality of correlations of levels with discrete statistics

TL;DR

This paper investigates the universal statistical properties of discrete level systems with logarithmic interactions, deriving closed-form correlators and analyzing their behavior in large and scaling limits.

Contribution

It introduces a generalized model with external sources and provides explicit formulas for correlators across different regimes, revealing universality in level correlations.

Findings

Closed-form correlators for any N

Universal correlation functions in Dyson's limit

Density of levels in large and double scaling limits

Abstract

We study the statistics of a system of N random levels with integer values, in the presence of a logarithmic repulsive potential of Dyson type. This probleme arises in sums over representations (Young tableaux) of GL(N) in various matrix problems and in the study of statistics of partitions for the permutation group. The model is generalized to include an external source and its correlators are found in closed form for any N. We reproduce the density of levels in the large N and double scaling limits and the universal correlation functions in Dyson's short-distance scaling limit. We also study the statistics of small levels.