Asymptotics of Jack measures with homogeneous specializations

2025/09/11 by Evgeni Dimitrov, Xiaohan Gao, Dimitrov, Evgeni +5 · 2 citations
Mathematics · #60F10 #82B05 #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2509.09814

openalex publication_date 2025/09/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider Jack measures on partitions with homogeneous defining specializations. For each of the six distinct classes of measures obtained this way we prove a global law of large numbers with an explicit limiting particle density. We also demonstrate for one of these classes how to obtain a global central limit theorem, global and edge large deviation principles, and edge universality using the results of the paper. Our argument is based on explicitly evaluating Jack symmetric functions at homogeneous specializations, relating the Jack measures to the discrete β-ensembles from (Publications math' ematiques de l'IH' ES 125, 1-78, 2017) and using the discrete loop equations to substantially reduce computations.

Citations

• Universality of global asymptotics of Jack-deformed random Young diagrams at varying temperatures
• Gaussian Asymptotics of Jack Measures on Partitions from Weighted Enumeration of Ribbon Paths
• Rigidity and Edge Universality of Discrete β-Ensembles
• Stein's Method, Jack Measure, and the Metropolis Algorithm
• Uniform Asymptotics for Polynomials Orthogonal With Respect to a General Class of Discrete Weights and Universality Results for Associated Ensembles
• Infinite wedge and random partitions
• Infinite wedge and random partitions
• On the Distribution of the Length of the Longest Increasing Subsequence of Random Permutations
• Anisotropic Young diagrams and Jack symmetric functions

Cited by

• Crystallization of discrete N-particle systems at high temperature
• Central limit theorem for the sine-β point process at β≤ 2

Related

• Stein's Method, Jack Measure, and the Metropolis Algorithm
• Universality of global asymptotics of Jack-deformed random Young diagrams at varying temperatures
• Gaussian Asymptotics of Jack Measures on Partitions from Weighted Enumeration of Ribbon Paths
• Law of Large Numbers and Central Limit Theorems by Jack Generating Functions
• A global large deviation principle for discrete β-ensembles