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

Panagiotis Zografos

TL;DR

This work introduces a family of $q$-deformed Perelomov-Popov measures $m_{N,PP(q)}$ on rescaled signatures and establishes a Law of Large Numbers to deterministic limits $oldsymbol{ u}^{(q)}$ with explicit moment and free cumulant formulas. The authors develop a differential-operator framework on Schur generating functions to extract moments, derive a $q$-deformed R-transform $R^{(q)}(z)=e_q(z)oldsymbol{ ext Ψ}'(e_q(z))+ rac{e_q(z)}{e_q(z)-1}- rac{1}{z}$, and reveal a smooth interpolation between $q=0$ and $q=1$, tying the results to free probability and its quantized variants. The paper further connects to infinitesimal free probability, providing explicit infinitesimal cumulants and R-transforms that describe first-order corrections, and uncovers non-asymptotic Markov-Krein-type correspondences (via maps between measures) that relate limiting laws across $q$. Throughout, extreme characters of $U( ext{infty})$ illuminate the limiting distributions (e.g., semicircle, Marchenko-Pastur) and their infinitesimal deformations, with concrete density formulas and potential links to random matrix ensembles. Overall, the work offers a comprehensive, $q$-parametrized bridge between representation-theoretic, combinatorial, and free-probabilistic perspectives on large-unitary-group asymptotics.

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.

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

TL;DR

This work introduces a family of -deformed Perelomov-Popov measures on rescaled signatures and establishes a Law of Large Numbers to deterministic limits with explicit moment and free cumulant formulas. The authors develop a differential-operator framework on Schur generating functions to extract moments, derive a -deformed R-transform , and reveal a smooth interpolation between and , tying the results to free probability and its quantized variants. The paper further connects to infinitesimal free probability, providing explicit infinitesimal cumulants and R-transforms that describe first-order corrections, and uncovers non-asymptotic Markov-Krein-type correspondences (via maps between measures) that relate limiting laws across . Throughout, extreme characters of illuminate the limiting distributions (e.g., semicircle, Marchenko-Pastur) and their infinitesimal deformations, with concrete density formulas and potential links to random matrix ensembles. Overall, the work offers a comprehensive, -parametrized bridge between representation-theoretic, combinatorial, and free-probabilistic perspectives on large-unitary-group asymptotics.

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 . 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 , 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 , which are related to the celebrated Markov-Krein correspondence.
Paper Structure (14 sections, 15 theorems, 148 equations, 1 figure)

This paper contains 14 sections, 15 theorems, 148 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview
  3. Free convolution and its quantized versions
  4. Related works
  5. Law of Large Numbers for Perelomov-Popov measures
  6. Schur generating functions and differential operators
  7. Asymptotic behavior and moments
  8. Asymptotic behavior and $R$-transform
  9. Extreme Characters and $q$-deformed Perelomov-Popov measures
  10. Perelomov-Popov measures and infinitesimal free probability
  11. Asymptotic behavior and infinitesimal moments
  12. Asymptotic behavior and infinitesimal $R$-transform
  13. Infinitesimal distributions and extreme characters
  14. Non-asymptotic relations and Markov-Krein correspondence

Key Result

Theorem 1

Let $\varrho (N)$, $N \in \mathbb{N}$, be a sequence of probability measures on $\hat{U} (N)$ such that for every $k \in \mathbb{N}$ and $i_0, \ldots, i_m \in \{1, \ldots, N\}$ with $|\{i_1, \ldots, i_m \}| \geq 2$. Moreover assume that the power series converges in a neighborhood of $1$. Then the sequence of random measures $m_{N, P P (q)} [\varrho (N)]$ converges as $N \rightarrow \infty$ in

Figures (1)

  • Figure 1: Graph of $y = \widetilde{f_{\gamma, q}} (x)$ for $\gamma = 0.25$ and different values of $q \in [- 1, 1]$.

Theorems & Definitions (29)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Definition 1: Speicher, B15
  • Proposition 1
  • Proposition 2
  • Remark 1
  • Proposition 3
  • Theorem 3
  • Definition 2
  • ...and 19 more