Asymptotics of Harish-Chandra transform and infinitesimal freeness

TL;DR

This paper extends the Harish-Chandra transform techniques to analyze asymptotic behaviors of particle systems and random matrices, revealing phase transitions and connecting with infinitesimal free probability.

Contribution

It introduces new general results for asymptotic expansions, links to infinitesimal free probability, and explores multiple growth regimes of the Harish-Chandra transform.

Findings

Established asymptotic expansion results for empirical measures.

Identified an analog of the Baik-Ben Arous-Peche phase transition.

Proved Law of Large Numbers in new growth regimes.

Abstract

In the last ten years a technique of Schur generating functions and Harish-Chandra transforms was developed for the study of the asymptotic behavior of discrete particle systems and random matrices. In the current paper we extend this toolbox in several directions. We establish general results which allow to access not only the Law of Large Numbers, but also next terms of the asymptotic expansion of averaged empirical measures. In particular, this allows to obtain an analog of a discrete Baik-Ben Arous-Peche phase transition. A connection with infinitesimal free probability is shown and a quantized version of infinitesimal free probability is introduced. Also, we establish the Law of Large Numbers for several new regimes of growth of a Harish-Chandra transform.