Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Random walks on regular trees can not be slowed down

Omer Angel, Jacob Richey, Yinon Spinka, Amir Yehudayoff

TL;DR

The paper investigates whether a random walk on a $d$-regular tree can be slowed by time-dependent vertex permutations that do not depend on the particle's location. It develops a speed-process framework anchored in majorization and sharp isoperimetric inequalities, complemented by a spectral-argument perspective, to compare permuted walks with ordinary lazy and non-lazy walks and to characterize exceptional times. The main findings show that for all $d\ge 2$ and any permutation sequence, the permuted lazy walk $|Y_t|$ stochastically dominates the ordinary lazy walk $|X_t|$, yielding $\liminf_{t\to\infty} t^{-1} |Y_t| \ge \frac{d-2}{d+1}$; for non-lazy walks a similar domination holds for $d>2$ (with $|Z_t|$ and $|S_t|$ giving a bound $\ge \frac{d-2}{d}$). The work also delineates exceptional times: such slowdown phenomena occur on $\mathbb Z$ but are absent for $d>2$ under automorphisms, with couplings showing gaps of order $t^{1/2-o(1)}$ in the latter case. Overall, the results reveal a rigidity of speed on regular trees under time-dependent relabellings and highlight the power of isoperimetric and majorization techniques in analyzing perturbations of Markov chains.

Abstract

A random walk on a regular tree (or any non-amenable graph) has positive speed. We ask whether such a walk can be slowed down by applying carefully chosen time-dependent permutations of the vertices. We prove that on trees the random walk can not be slowed down.

Random walks on regular trees can not be slowed down

TL;DR

The paper investigates whether a random walk on a -regular tree can be slowed by time-dependent vertex permutations that do not depend on the particle's location. It develops a speed-process framework anchored in majorization and sharp isoperimetric inequalities, complemented by a spectral-argument perspective, to compare permuted walks with ordinary lazy and non-lazy walks and to characterize exceptional times. The main findings show that for all and any permutation sequence, the permuted lazy walk stochastically dominates the ordinary lazy walk , yielding ; for non-lazy walks a similar domination holds for (with and giving a bound ). The work also delineates exceptional times: such slowdown phenomena occur on but are absent for under automorphisms, with couplings showing gaps of order in the latter case. Overall, the results reveal a rigidity of speed on regular trees under time-dependent relabellings and highlight the power of isoperimetric and majorization techniques in analyzing perturbations of Markov chains.

Abstract

A random walk on a regular tree (or any non-amenable graph) has positive speed. We ask whether such a walk can be slowed down by applying carefully chosen time-dependent permutations of the vertices. We prove that on trees the random walk can not be slowed down.
Paper Structure (11 sections, 16 theorems, 55 equations)

This paper contains 11 sections, 16 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Permuted random walks
  3. Non-lazy random walks
  4. The speed process
  5. A spectral argument
  6. Isoperimetry
  7. Lazy random walks
  8. Simple random walks
  9. Exceptional times
  10. Exceptional times for $\mathbb Z$
  11. No exceptional times for $d>2$

Key Result

Theorem 1.1

For every $d \geq 2$, every sequence $(\pi_t)$ of permutations of $V(\mathbb T_d)$, and every time $t \ge 0$, the depth of the permuted random walk $|Y_t|$ stochastically dominates the depth of the lazy random walk $|X_t|$. That is, for all $t,n \geq 0$, In particular, $\mathbb E |Y_t| \geq \mathbb E |X_t|$ for all $t \geq 0$, and $\liminf_{t \to \infty} t^{-1} |Y_t| \geq \frac{d-2}{d+1}$ almost

Theorems & Definitions (35)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Proposition 2.1
  • Proposition 2.2
  • ...and 25 more