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Discrete $N$-particle systems at high temperature through Jack generating functions

Cesar Cuenca, Maciej Dołęga

TL;DR

This work develops a comprehensive framework for the Law of Large Numbers for discrete N-particle systems at high temperature, expressed through Jack generating functions. By analyzing the action of Cherednik operators and exploiting Hecke relations, the authors derive necessary and sufficient HT-appropriateness conditions that encode the limiting cumulants κ_n^γ and the limiting moments m_ℓ via a Lukasiewicz-path transform. They prove that HT-appropriateness is equivalent to LLN satisfaction and provide an explicit inverse transform between cumulants and moments, enabling a combinatorial description of limiting distributions μ^γ. The theory yields concrete applications to Jack measures and nonintersecting particle systems, including a γ-deformed convolution framework and LLNs for pure alpha/beta/plancherel Jack measures, and it reconciles with prior fixed-temperature results while highlighting new discrete high-temperature phenomena and connections to Stanley’s conjectures on Jack LR coefficients.

Abstract

We find necessary and sufficient conditions for the Law of Large Numbers for random discrete $N$-particle systems with the deformation (inverse temperature) parameter $θ$, as their size $N$ tends to infinity simultaneously with the inverse temperature going to zero. Our conditions are expressed in terms of the Jack generating functions, and our analysis is based on the asymptotics of the action of Cherednik operators obtained via Hecke relations. We apply the general framework to obtain the LLN for a large class of Markov chains of $N$ nonintersecting particles with interaction of log-gas type, and the LLN for the multiplication of Jack polynomials, as the inverse temperature tends to zero. We express the answer in terms of novel one-parameter deformations of cumulants and their description provided by us recovers previous work by Bufetov--Gorin on quantized free cumulants when $θ=1$, and by Benaych-Georges--Cuenca--Gorin after a deformation to continuous space of random matrix eigenvalues. Our methods are robust enough to be applied to the fixed temperature regime, where we recover the LLN of Huang.

Discrete $N$-particle systems at high temperature through Jack generating functions

TL;DR

This work develops a comprehensive framework for the Law of Large Numbers for discrete N-particle systems at high temperature, expressed through Jack generating functions. By analyzing the action of Cherednik operators and exploiting Hecke relations, the authors derive necessary and sufficient HT-appropriateness conditions that encode the limiting cumulants κ_n^γ and the limiting moments m_ℓ via a Lukasiewicz-path transform. They prove that HT-appropriateness is equivalent to LLN satisfaction and provide an explicit inverse transform between cumulants and moments, enabling a combinatorial description of limiting distributions μ^γ. The theory yields concrete applications to Jack measures and nonintersecting particle systems, including a γ-deformed convolution framework and LLNs for pure alpha/beta/plancherel Jack measures, and it reconciles with prior fixed-temperature results while highlighting new discrete high-temperature phenomena and connections to Stanley’s conjectures on Jack LR coefficients.

Abstract

We find necessary and sufficient conditions for the Law of Large Numbers for random discrete -particle systems with the deformation (inverse temperature) parameter , as their size tends to infinity simultaneously with the inverse temperature going to zero. Our conditions are expressed in terms of the Jack generating functions, and our analysis is based on the asymptotics of the action of Cherednik operators obtained via Hecke relations. We apply the general framework to obtain the LLN for a large class of Markov chains of nonintersecting particles with interaction of log-gas type, and the LLN for the multiplication of Jack polynomials, as the inverse temperature tends to zero. We express the answer in terms of novel one-parameter deformations of cumulants and their description provided by us recovers previous work by Bufetov--Gorin on quantized free cumulants when , and by Benaych-Georges--Cuenca--Gorin after a deformation to continuous space of random matrix eigenvalues. Our methods are robust enough to be applied to the fixed temperature regime, where we recover the LLN of Huang.

Paper Structure

This paper contains 32 sections, 29 theorems, 207 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Main results
  4. Applications
  5. Quantized $\gamma$-convolution and the Jack Littlewood--Richardson coefficients
  6. Nonintersecting particle systems in the high temperature regime through Jack measures
  7. Previous related works
  8. Methodology
  9. Jack polynomial theory and Jack generating functions
  10. Partitions and signatures
  11. The Jack parameter
  12. Jack Laurent polynomials
  13. Cherednik operators
  14. Measures on $N$-signatures and Jack generating functions
  15. Jack symmetric functions
  16. ...and 17 more sections

Key Result

Theorem 1

Let $\{\mathbb{P}_N\}_{N\ge 1}$ be a sequence of finite signed measures on $N$-signatures (that satisfies a natural condition of def:small_tails on the growth rate of $G_{\mathbb P_N,\theta}$) and let $\{\mu_N\}_{N\ge 1}$ be their empirical measures. Then $\mathbb{E}_{\mathbb{P}_N}\mu_N$ converge in if and only if the following two conditions are satisfied, for some constants $\kappa_1^\gamma,\kap

Figures (2)

  • Figure 1: Three simulations of the Gorin--Shkolnikov process with $N=60$ particles and initial configuration $\mathscr{L}^{(0)}_i = \theta(1-i)$, for $1 \leq i \leq N$. The simulations show the asymptotic behavior in the fixed temperature regime on the left (with $\theta=1$), and in the high temperature regime with $\gamma = 2$ in the middle, and with $\gamma = \frac{1}{2}$ on the right. The $x$-axis represents the space and the $y$-axis represents the time.
  • Figure 2: There are five Łukasiewicz paths of length $3$. We present them with the weights associated to each of their steps, and the overall contribution after applying the divided difference operator $\Delta_\gamma$. The horizontal steps at height $0$ and their weight after applying the divided difference operator are marked in blue.

Theorems & Definitions (76)

  • Theorem 1: First part of \ref{['theo:main1']} in the text
  • Theorem 2: Second part of \ref{['theo:main1']} in the text
  • Theorem 3: \ref{['theo:LLN_Markov']} in the text
  • Theorem 4
  • Definition 1
  • Example 1
  • Remark 1
  • Definition 2
  • Example 2
  • Example 3
  • ...and 66 more