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On the return probability of the simple random walk on Galton-Watson trees

Peter Müller, Jakob Stern

TL;DR

This work analyzes the annealed return probability $R_t$ of a simple random walk on a Galton--Watson tree conditioned on non-extinction. By extending Virág's anchored-expansion approach and introducing probabilistic control of bad-tree geometries alongside a time-dependent regularization, the authors construct an induced Markov chain on the $q$-oceans that inherits favorable isoperimetric properties. They prove a sharp subexponential bound with optimal exponent $1/3$ for bounded offspring support, and extend the bounds to unbounded offspring with tails decaying as $p_j\le c_1 e^{-c_2 j^k}$, yielding $R_t \le \exp(-c t^{1/3-2/(3k)})$, with further improvements for faster decays. The results connect geometric properties of random trees to spectral bounds for random walks, yielding precise annealed heat-kernel decay rates and contributing to understanding of random-graph limits and spectral theory on trees.

Abstract

We consider the simple random walk on Galton-Watson trees with supercritical offspring distribution, conditioned on non-extinction. In case the offspring distribution has finite support, we prove an upper bound for the annealed return probability to the root which decays subexponentially in time with exponent 1/3. This exponent is optimal. Our result improves the previously known subexponential upper bound with exponent 1/5 by Piau [Ann. Probab. 26, 1016-1040 (1998)]. For offspring distributions with unbounded support but sufficiently fast decay, our method also yields improved subexponential upper bounds.

On the return probability of the simple random walk on Galton-Watson trees

TL;DR

This work analyzes the annealed return probability of a simple random walk on a Galton--Watson tree conditioned on non-extinction. By extending Virág's anchored-expansion approach and introducing probabilistic control of bad-tree geometries alongside a time-dependent regularization, the authors construct an induced Markov chain on the -oceans that inherits favorable isoperimetric properties. They prove a sharp subexponential bound with optimal exponent for bounded offspring support, and extend the bounds to unbounded offspring with tails decaying as , yielding , with further improvements for faster decays. The results connect geometric properties of random trees to spectral bounds for random walks, yielding precise annealed heat-kernel decay rates and contributing to understanding of random-graph limits and spectral theory on trees.

Abstract

We consider the simple random walk on Galton-Watson trees with supercritical offspring distribution, conditioned on non-extinction. In case the offspring distribution has finite support, we prove an upper bound for the annealed return probability to the root which decays subexponentially in time with exponent 1/3. This exponent is optimal. Our result improves the previously known subexponential upper bound with exponent 1/5 by Piau [Ann. Probab. 26, 1016-1040 (1998)]. For offspring distributions with unbounded support but sufficiently fast decay, our method also yields improved subexponential upper bounds.
Paper Structure (5 sections, 18 theorems, 97 equations, 1 table)

This paper contains 5 sections, 18 theorems, 97 equations, 1 table.

Table of Contents

  1. Introduction and result
  2. Anchored expansion and events of bad trees
  3. Markov-kernel estimates for an effective random walk
  4. Proof of Theorems \ref{['Decay']} and \ref{['thm']}
  5. On q-islands and q-oceans

Key Result

Theorem 1.2

Let $\mathds{T}$ be a supercritical Galton--Watson tree conditioned on non-extinction and assume that its offspring distribution has bounded support. Then, there exists a constant $c>0$ such that for all $t\in\mathds{N}_{0}$ we have

Theorems & Definitions (46)

  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Corollary 1.5
  • proof
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3
  • ...and 36 more