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A central limit theorem for a generalization of the Ewens measure to random tuples of commuting permutations

Abdelmalek Abdesselam, Shannon Starr

TL;DR

The paper studies the number of joint orbits in random tuples of pairwise commuting permutations and proves a central limit theorem for this orbit count under an Ewens-like weighting on tuples. It introduces a generating-function framework via the Bryan–Fulman product and performs a detailed saddle-point analysis to control the coefficient asymptotics, splitting the contour into major and minor arcs. The authors derive explicit asymptotics for the mean and variance, $\mathbb{E}\mathsf{K}_{\ell,n}$ and ${\rm Var}(\mathsf{K}_{\ell,n})$, and establish that the normalized joint-orbit count converges in distribution to a standard Gaussian for all $\ell\ge 2$, with a logarithmic correction in the classical $\ell=1$ case. The approach connects probabilistic limit theorems with number-theoretic tools (zeta functions and Meinardus-type poles) and a combinatorial framework (Ewens-like measures) for random commuting tuples, providing a self-contained treatment that also suggests avenues toward local limit theorems, large deviations, and probabilistic couplings in random commuting matrix models. The results extend Goncharov’s CLT to higher-dimensional commuting structures and tie into broader themes in random combinatorial structures and discrete matrix models.

Abstract

We prove a central limit theorem (CLT) for the number of joint orbits of random tuples of commuting permutations. In the uniform sampling case this generalizes the classic CLT of Goncharov for the number of cycles of a single random permutation. We also consider the case where tuples are weighted by a factor other than one, per joint orbit. We view this as an analogue of the Ewens measure, for tuples of commuting permutations, where our CLT generalizes the CLT by Hansen. Our proof uses saddle point analysis, in a context related to the Hardy-Ramanujan asymptotics and the theorem of Meinardus, but concerns a multiple pole situation. The proof is written in a self-contained manner, and hopefully in a manner accessible to a wider audience. We also indicate several open directions of further study related to probability, combinatorics, number theory, an elusive theory of random commuting matrices, and perhaps also geometric group theory.

A central limit theorem for a generalization of the Ewens measure to random tuples of commuting permutations

TL;DR

The paper studies the number of joint orbits in random tuples of pairwise commuting permutations and proves a central limit theorem for this orbit count under an Ewens-like weighting on tuples. It introduces a generating-function framework via the Bryan–Fulman product and performs a detailed saddle-point analysis to control the coefficient asymptotics, splitting the contour into major and minor arcs. The authors derive explicit asymptotics for the mean and variance, and , and establish that the normalized joint-orbit count converges in distribution to a standard Gaussian for all , with a logarithmic correction in the classical case. The approach connects probabilistic limit theorems with number-theoretic tools (zeta functions and Meinardus-type poles) and a combinatorial framework (Ewens-like measures) for random commuting tuples, providing a self-contained treatment that also suggests avenues toward local limit theorems, large deviations, and probabilistic couplings in random commuting matrix models. The results extend Goncharov’s CLT to higher-dimensional commuting structures and tie into broader themes in random combinatorial structures and discrete matrix models.

Abstract

We prove a central limit theorem (CLT) for the number of joint orbits of random tuples of commuting permutations. In the uniform sampling case this generalizes the classic CLT of Goncharov for the number of cycles of a single random permutation. We also consider the case where tuples are weighted by a factor other than one, per joint orbit. We view this as an analogue of the Ewens measure, for tuples of commuting permutations, where our CLT generalizes the CLT by Hansen. Our proof uses saddle point analysis, in a context related to the Hardy-Ramanujan asymptotics and the theorem of Meinardus, but concerns a multiple pole situation. The proof is written in a self-contained manner, and hopefully in a manner accessible to a wider audience. We also indicate several open directions of further study related to probability, combinatorics, number theory, an elusive theory of random commuting matrices, and perhaps also geometric group theory.
Paper Structure (11 sections, 8 theorems, 117 equations)

This paper contains 11 sections, 8 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. The main result
  3. A brief outline of the proof
  4. Relation to other work and possible directions for further inquiry
  5. Preliminaries on the $Z$ functions
  6. Preparation for the saddle point analysis
  7. The saddle point analysis
  8. Major arc estimates
  9. Minor arc estimates
  10. Putting the estimates together and deriving the $\mathcal{J}$ integral asymptotics
  11. Completion of the proofs of the main proposition and theorem

Key Result

Theorem 1.1

For any $\ell\ge 2$, and any $x>0$, as $n\rightarrow\infty$, the leading asymptotics of the mean and variance of the random variables $\mathsf{K}_{\ell,n}$ are given by Moreover, the normalized random variables converge in distribution and in the sense of moments to the standard Gaussian $\mathcal{N}(0,1)$. Namely, we have for all $f$'s that are bounded continuous functions, or polynomials.

Theorems & Definitions (8)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.1
  • Proposition 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4