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

Cesar Cuenca, Jiaming Xu

TL;DR

This work establishes a complete fixed-temperature Law of Large Numbers for averaged empirical measures of $N$-particle ensembles by linking LLN to the asymptotics of Bessel generating functions $G_{N,\theta}$. It provides necessary and sufficient conditions (the $\theta$-LLN-appropriate criteria) and expresses the limiting moments $m_k$ through cumulants $\kappa_d$ via Lukasiewicz-path combinatorics, bridging probabilistic limits with a topological expansion. The authors develop both Dunkl-operator methods (for the 'if' direction) and Chapuy–Dołęga's constellation expansion (for the 'only if' direction), yielding a robust framework that yields LLN results for $\theta$-sums, $\theta$-corners, and time-slices of the $\theta$-Dyson Brownian motion. Applications recover and extend free-convolution phenomena in fixed-temperature deformations and provide a geometric–combinatorial interpretation of the cumulant-moment relations. Overall, the paper advances the analytic- combinatorial toolbox for linking random-matrix ensembles with free probability at fixed temperature.

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 $θ$-sums and $θ$-corners of random matrices are given by the free convolution and free projection, respectively, regardless of the value of inverse temperature parameter $θ$. We also prove the LLN for a time-slice of the $θ$-Dyson Brownian motion.

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

TL;DR

This work establishes a complete fixed-temperature Law of Large Numbers for averaged empirical measures of -particle ensembles by linking LLN to the asymptotics of Bessel generating functions . It provides necessary and sufficient conditions (the -LLN-appropriate criteria) and expresses the limiting moments through cumulants via Lukasiewicz-path combinatorics, bridging probabilistic limits with a topological expansion. The authors develop both Dunkl-operator methods (for the 'if' direction) and Chapuy–Dołęga's constellation expansion (for the 'only if' direction), yielding a robust framework that yields LLN results for -sums, -corners, and time-slices of the -Dyson Brownian motion. Applications recover and extend free-convolution phenomena in fixed-temperature deformations and provide a geometric–combinatorial interpretation of the cumulant-moment relations. Overall, the paper advances the analytic- combinatorial toolbox for linking random-matrix ensembles with free probability at fixed temperature.

Abstract

In this paper, we find necessary and sufficient conditions for the Law of Large Numbers of averaged empirical measures of -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 -sums and -corners of random matrices are given by the free convolution and free projection, respectively, regardless of the value of inverse temperature parameter . We also prove the LLN for a time-slice of the -Dyson Brownian motion.

Paper Structure

This paper contains 22 sections, 14 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Main result
  4. Applications: LLN for $\theta$-sums and $\theta$-corners
  5. Methodology
  6. Bessel generating functions
  7. Conventions
  8. Dunkl operators
  9. Multivariate Bessel functions
  10. Bessel generating functions
  11. Statement of the main theorem
  12. Sufficient conditions for the LLN
  13. Technical lemmas on the action of Dunkl operators
  14. Proof of the "if" part of the main theorem
  15. Necessary conditions for the LLN
  16. ...and 7 more sections

Key Result

Theorem 1.1

The following two statements are equivalent, regarding a sequence $\{\mu_N\}_{N\ge 1}$, where each $\mu_N$ is a probability measure on $\mathbb R^N$. If one (and therefore both) of the previous two equivalent statements holds, then $\{m_k\}_{k\ge 1}$ and $\{\kappa_\ell\}_{\ell\ge 1}$ are related to each other by the moment -- free cumulant formulas in equation eq_fixedtempmoment-cumulant.

Theorems & Definitions (35)

  • Theorem 1.1: See Theorem \ref{['thm:main']} in the text for details
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 2.1: O
  • Proposition 2.3: Prop. 2.11 from BCG
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.4: Main Theorem
  • ...and 25 more