Law of Large Numbers for continuous $N$-particle ensembles at fixed temperature

TL;DR

This paper establishes necessary and sufficient conditions for the Law of Large Numbers in N-particle ensembles at fixed temperature, using Bessel generating functions and Dunkl operators, with applications to random matrices and Dyson Brownian motion.

Contribution

It provides a complete characterization of LLN conditions for particle ensembles at fixed temperature, solving an open problem and connecting to free probability.

Findings

LLN for $ heta$-sums and $ heta$-corners corresponds to free convolution and free projection.

Proves LLN for a time-slice of $ heta$-Dyson Brownian motion.

Conditions are expressed via asymptotics of Bessel generating functions.

Abstract

In this paper, we find necessary and sufficient conditions for the Law of Large Numbers of averaged empirical measures of $N$-particle ensembles, in terms of the asymptotics of their Bessel generating functions, in the fixed temperature regime. This settles an open problem posed by Benaych-Georges, Cuenca and Gorin. For one direction, we use the moment method through Dunkl operators, and for the other we employ a special case of the formula of Chapuy--Dolega for the generating function of infinite constellations. As applications, we prove that the LLN for $\theta$-sums and $\theta$-corners of random matrices are given by the free convolution and free projection, respectively, regardless of the value of inverse temperature parameter $\theta$. We also prove the LLN for a time-slice of the $\theta$-Dyson Brownian motion.