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Traces left by random walks on large random graphs: local limits

Steffen Dereich

Abstract

In this article, we develop a theory for understanding the traces left by a random walk in the vicinity of a randomly chosen reference vertex. The analysis is related to interlacements but goes beyond previous research by showing weak limit theorems for the vicinity around the reference vertex together with the path segments of the random walk in this vicinity. Roughly speaking, our limit theorem requires appropriate mixing properties for the random walk together with a stronger variant of Benjamini-Schramm convergence for the underlying graph model. If these assumptions are satisfied, the limiting object can be explicitly given in terms of the Benjamini-Schramm limit of the graph model.

Traces left by random walks on large random graphs: local limits

Abstract

In this article, we develop a theory for understanding the traces left by a random walk in the vicinity of a randomly chosen reference vertex. The analysis is related to interlacements but goes beyond previous research by showing weak limit theorems for the vicinity around the reference vertex together with the path segments of the random walk in this vicinity. Roughly speaking, our limit theorem requires appropriate mixing properties for the random walk together with a stronger variant of Benjamini-Schramm convergence for the underlying graph model. If these assumptions are satisfied, the limiting object can be explicitly given in terms of the Benjamini-Schramm limit of the graph model.
Paper Structure (9 sections, 16 theorems, 176 equations)

This paper contains 9 sections, 16 theorems, 176 equations.

Table of Contents

  1. Introduction
  2. The setup
  3. The visiting counting measure
  4. The main result
  5. Exploration convergence
  6. Markov process estimates
  7. Proof of the main result
  8. Vacant set percolation
  9. Exploration convergence for the configuration model

Key Result

theorem 1

Let $(\ell_n')_{n\in\mathbb N}$ and $(\ell_n)_{n\in\mathbb N}$ be $\mathbb N$-valued and $(\tau_n)_{n\in\mathbb N}$ and $(a_n)_{n\in\mathbb N}$ be $(0,\infty)$-valued with all three sequences tending to infinity and suppose that the random finite connected $\mathbb{G}^\mathrm{root}$-graphs $G_1^{o_1 Let $R,R'\in\mathbb N$ and $\sigma>0$. For $n\in\mathbb N$, let $\Xi_n$ be the $(V_n^{o_n}[R],\tau_

Theorems & Definitions (43)

  • theorem 1: Coupling visiting measures
  • Remark 1.1: Interlacement representation
  • Example 1.2: Random graphs with fixed degree sequence
  • Example 1.3: Erdős-Rényi random graphs
  • Definition 1.4
  • Example 1.5: Breadth-first exploration
  • Example 1.6: Exploration governed by the Markov procecss
  • Definition 1.7
  • proposition 1
  • Remark 1.8
  • ...and 33 more