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Return probability on Bienaymé-Galton-Watson trees and spectral asymptotics of sparse Erdős-Rényi random graphs

Markus Heydenreich, Peter Müller, Sara Terveer

Abstract

We derive an upper bound for the annealed return probability for the simple random walk on supercritical Bienaymé-Galton-Watson trees. The bound decays subexponentially in time $t$ with $t^{1/3}$ in the exponent. It is valid for all offspring distributions with a finite first moment and is optimal whenever the offspring distribution does not exclude leaves or linear pieces in the tree. This solves completely the case left open by Piau [Ann. Probab. 26, 1016-1040 (1998)]. In the special case of a Poissonian offspring distribution we apply this upper bound to deduce a Lifshits tail for the empirical eigenvalue distribution of the graph Laplacian on supercritical Erdős-Rényi random graphs with finite mean degree.

Return probability on Bienaymé-Galton-Watson trees and spectral asymptotics of sparse Erdős-Rényi random graphs

Abstract

We derive an upper bound for the annealed return probability for the simple random walk on supercritical Bienaymé-Galton-Watson trees. The bound decays subexponentially in time with in the exponent. It is valid for all offspring distributions with a finite first moment and is optimal whenever the offspring distribution does not exclude leaves or linear pieces in the tree. This solves completely the case left open by Piau [Ann. Probab. 26, 1016-1040 (1998)]. In the special case of a Poissonian offspring distribution we apply this upper bound to deduce a Lifshits tail for the empirical eigenvalue distribution of the graph Laplacian on supercritical Erdős-Rényi random graphs with finite mean degree.
Paper Structure (13 sections, 13 theorems, 75 equations)

This paper contains 13 sections, 13 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Return probability on Bienaymé--Galton--Watson trees
  3. Empirical eigenvalue distribution of the Laplacian on finite-mean-degree Erdős--Rényi random graphs
  4. Return probabilities of continuous-time random walks
  5. Organisation of the paper
  6. Properties of BGWTs conditioned on survival
  7. Isoperimetric ratios and consequences
  8. Isolated cores
  9. Skipping islands
  10. Useful consequences of anchored expansion
  11. Proof of Theorem \ref{['thm:returnprobability']}
  12. Proof of Theorem \ref{['thm:lifshits']}
  13. Proof of Corollary \ref{['cor:cont-time-RW']}

Key Result

Theorem 1.1

Assume that the first moment $\lambda$ of the offspring distribution $\mu$ exists and satisfies $\lambda>1$. Then, there exists a constant $c>0$ such that for every $t\in\mathbb{N}_0$.

Theorems & Definitions (28)

  • Theorem 1.1
  • Lemma 1.2
  • Theorem 1.3
  • Corollary 1.5
  • Remark 1.6
  • Definition 3.1
  • Remark 3.2
  • Lemma 3.3
  • Definition 3.4
  • Definition 3.5
  • ...and 18 more