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Cutoff for random Cayley graphs of nilpotent groups

Jonathan Hermon, Xiangying Huang

Abstract

We consider the random Cayley graphs of a sequence of finite nilpotent groups of diverging sizes $G=G(n)$, whose ranks and nilpotency classes are uniformly bounded. For some $k=k(n)$ such that $1\ll\log k \ll \log |G|$, we pick a random set of generators $S=S(n)$ by sampling $k$ elements $Z_1,\ldots,Z_k$ from $G$ uniformly at random with replacement, and set $S:=\{Z_j^{\pm 1}:1 \le j\le k \}$. We show that the simple random walk on Cay$(G,S)$ exhibits cutoff with high probability. Some of our results apply to a general set of generators. Namely, we show that there is a constant $c>0$, depending only on the rank and the nilpotency class of $G$, such that for all symmetric sets of generators $S$ of size at most $ \frac{c\log |G|}{\log \log |G|}$, the spectral gap and the $\varepsilon$-mixing time of the simple random walk $X=(X_t)_{t\geq 0}$ on Cay$(G,S)$ are asymptotically the same as those of the projection of $X$ to the abelianization of $G$, given by $[G,G]X_t$. In particular, $X$ exhibits cutoff if and only if its projection does.

Cutoff for random Cayley graphs of nilpotent groups

Abstract

We consider the random Cayley graphs of a sequence of finite nilpotent groups of diverging sizes , whose ranks and nilpotency classes are uniformly bounded. For some such that , we pick a random set of generators by sampling elements from uniformly at random with replacement, and set . We show that the simple random walk on Cay exhibits cutoff with high probability. Some of our results apply to a general set of generators. Namely, we show that there is a constant , depending only on the rank and the nilpotency class of , such that for all symmetric sets of generators of size at most , the spectral gap and the -mixing time of the simple random walk on Cay are asymptotically the same as those of the projection of to the abelianization of , given by . In particular, exhibits cutoff if and only if its projection does.
Paper Structure (37 sections, 46 theorems, 286 equations)

This paper contains 37 sections, 46 theorems, 286 equations.

Table of Contents

  1. Introduction
  2. Motivation and Objectives of Paper
  3. Motivations
  4. Objectives
  5. Definitions and Notation
  6. Overview of Main Results
  7. Cutoff for Random Walks on Nilpotent Groups
  8. Random Walk on Non-random Cayley Graphs: Reduction to Abelianization
  9. Our Methodology
  10. Historic Overview
  11. Random Walks on Unipotent Matrix Groups
  12. The Entropic Methodology
  13. Geometry of Cayley Graphs
  14. Preliminaries
  15. Proof of Theorem \ref{['diam_bound']}
  16. ...and 22 more sections

Key Result

Theorem 1

Let $G$ be a finite nilpotent group with $r(G), L(G)\asymp 1$. Let $S=\{ Z_i^{\pm 1}: i\in [k]\}$ with $Z_1,\dots,Z_k\overset{iid}{\sim} \mathrm{Unif}(G)$. Assume $1\ll \log k \ll \log|G|$. As $|G|\to \infty$, the random walk on $\text{Cay}(G,S)$ exhibits cutoff with high probability at time $t_*(k,

Theorems & Definitions (89)

  • Conjecture : Aldous and Diaconis, aldous1986shuffling
  • Definition 1: Cayley multi-graph generated by a set of generators
  • Remark 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Theorem 2
  • Remark 2
  • Corollary 1
  • Theorem 3
  • ...and 79 more