Classical beta ensembles and related eigenvalues processes at high temperature and the Markov--Krein transform

TL;DR

This paper investigates the high temperature limits of classical beta ensembles and related eigenvalue processes, revealing their measures as inverse Markov--Krein transforms of well-known distributions, and characterizing their limiting processes.

Contribution

It introduces the use of the Markov--Krein transform to identify the limits of beta ensembles and eigenvalue processes at high temperature, providing explicit descriptions of their limiting measures and processes.

Findings

Limiting measures are inverse Markov--Krein transforms of classical distributions.

The limiting eigenvalue processes are characterized as inverse Markov--Krein transforms of specific stochastic processes.

The results unify the understanding of high temperature regimes for various beta ensembles.

Abstract

The aim of this paper is to identify the limit in a high temperature regime of classical beta ensembles on the real line and related eigenvalue processes by using the Markov--Krein transform. We show that the limiting measure of Gaussian beta ensembles (resp.\ beta Laguerre ensembles and beta Jacobi ensembles) is the inverse Markov--Krein transform of the Gaussian distribution (resp.\ the gamma distribution and the beta distribution). At the process level, we show that the limiting probability measure-valued process is the inverse Markov--Krein transform of a certain 1d stochastic process.