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Poisson approximation for large permutation groups

Persi Diaconis, Nathan Tung

TL;DR

The paper extends classical Poisson limit results for random permutations to large wreath-product subgroups of $S_{kn}$ by studying the cycle structure of random elements in groups of the form $\\Gamma^n \\rtimes H$. It combines cycle-index theory, Poissonization/de-Poissonization, and coupling techniques to prove limit theorems in which the cycle counts $a_i(\\sigma)$ converge to joint compound Poisson distributions; the dependence structure among components is explicitly characterized in key cases. In particular, for $\\sigma \\in S_3^n \\rtimes S_n$, the limiting counts $A_i$ are compound Poisson with shared components, while for $\\sigma \\in C_k^n \\rtimes S_n$ the $A_i$ are independent, illustrating how internal block structure governs dependence. The results yield refined asymptotics for fixed points, cycles of various lengths, inversions, and related statistics, thereby generalizing classical permutation results to a broad family of structured groups with potential for applications in combinatorics and random partition generation.

Abstract

Let $G_{k,n}$ be a group of permutations of $kn$ objects which permutes things independently in disjoint blocks of size $k$ and then permutes the blocks. We investigate the probabilistic and/or enumerative aspects of random elements of $G_{k,n}$. This includes novel limit theorems for fixed points, cycles of various lengths, number of cycles and inversions. The limits are compound Poisson distributions with interesting dependence structure.

Poisson approximation for large permutation groups

TL;DR

The paper extends classical Poisson limit results for random permutations to large wreath-product subgroups of by studying the cycle structure of random elements in groups of the form . It combines cycle-index theory, Poissonization/de-Poissonization, and coupling techniques to prove limit theorems in which the cycle counts converge to joint compound Poisson distributions; the dependence structure among components is explicitly characterized in key cases. In particular, for , the limiting counts are compound Poisson with shared components, while for the are independent, illustrating how internal block structure governs dependence. The results yield refined asymptotics for fixed points, cycles of various lengths, inversions, and related statistics, thereby generalizing classical permutation results to a broad family of structured groups with potential for applications in combinatorics and random partition generation.

Abstract

Let be a group of permutations of objects which permutes things independently in disjoint blocks of size and then permutes the blocks. We investigate the probabilistic and/or enumerative aspects of random elements of . This includes novel limit theorems for fixed points, cycles of various lengths, number of cycles and inversions. The limits are compound Poisson distributions with interesting dependence structure.
Paper Structure (4 sections, 3 theorems, 6 equations)

This paper contains 4 sections, 3 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Wreath products
  4. The commuting graph process and random partitions

Key Result

Theorem 1

Pick $\sigma\in S_3^n\rtimes S_n$ uniformly. Write $\sigma\sim\prod_{i=1}^{3n}i^{a_i(\sigma)}$. Then $\{a_i(\sigma)\}_{i=1}^{3n}$ converges in distribution to $\{A_i\}_{i=1}^\infty$ with joint distribution where and all $W_i$, $Z_i$, $Y_i$ are independent.

Theorems & Definitions (5)

  • Example 1.1: $S_3^n\rtimes S_n$
  • Theorem 1
  • Example 1.2: $C_k^n\rtimes S_n$
  • Theorem 2
  • Proposition 1