$q$-deformed Perelomov-Popov measures and quantized free probability

TL;DR

This paper introduces a new family of $q$-deformed measures related to asymptotic representation theory, establishing laws of large numbers, explicit formulas for moments and free cumulants, and connecting to free probability and infinitesimal free probability.

Contribution

It develops a $q$-deformation of measures and free convolution, bridging results for $q=0,1$ and exploring their probabilistic and combinatorial properties.

Findings

Proved Law of Large Numbers for the new measures.

Derived explicit formulas for moments and free cumulants.

Established connections with free probability and infinitesimal free probability.

Abstract

The asymptotic study of tuples of random non-increasing integers is crucial for probabilistic models coming from asymptotic representation theory and statistical physics. We study the global behavior of such tuples, introducing a new family of discrete probability measures, depending on a parameter $q \in [- 1, 1]$. We prove the Law of Large Numbers for these measures based on the asymptotics of the Schur generating functions and we provide explicit formulas for the moments and the free cumulants of the limiting measures. Our results provide an interpolation between the results of Bufetov and Gorin for $q = 0, 1$, who distinguished these two cases from the side of free probability theory. We show the connection with free probability theory and we introduce a deformation of free convolution, motivated by our formulas for the free cumulants. We also study the first order correction to the Law of Large Numbers and we make the connection with infinitesimal free probability, computing explicitly the infinitesimal moments and the infinitesimal free cumulants. Finally, we prove non-asymptotic relations between the limiting measures for different $q$, which are related to the celebrated Markov-Krein correspondence.