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How thin does random interlacement have to be so that a random walk can see through it?

Nicolas Bouchot

TL;DR

The paper identifies a sharp threshold for when holes in the random interlacement trace become visible to a distant random walk, quantified by the window $\Theta_N$ and the scale $u_N$. It proves that the capacity ratio $\varsigma^{\mathrm{RI}}_{N,u_N}$ converges to $0$ or $1$ according as $u_N\Theta_N$ tends to $0$ or $+\infty$, with a nontrivial regime when the limit is finite; the threshold depends on dimension via $\Theta_N$. It then derives an analogous dichotomy for a simple random walk confined to a large ball up to time $t_N$, showing that the ratio of the walk’s range capacity to the ball’s capacity vanishes if $t_N\Theta_N/N^d\to 0$ and tends to 1 if $t_N\Theta_N/N^d\to\infty$, with dimension-dependent refinements. The proofs hinge on capacity decompositions, harmonic/equilibrium measures, and a key proposition controlling the capacity of the random-walk range, complemented by spectral-analytic arguments for the confined-walk setting and a dimension-3 excursion-based approach. Overall, the results illuminate how thinning random obstacles interact with random walks and reveal a concrete, dimension-sensitive phase transition in the visibility of holes through random interlacements.

Abstract

The random interlacements $\mathscr{I}(u)$ at level $u$ has been introduced by Sznitman, as a Poissonian collection of independent simple random walk trajectories on $\mathbb{Z}^d$, $d\geq 3$, with intensity $u>0$. Since then, several works investigated the properties of the random interlacements intersected with large sets of~$\mathbb{Z}^d$. In this paper, we study the asymptotic behavior of the capacity of $\mathscr{I}(u) \cap D_N$, where $D_N$ is the blow up of a compact set $D$, with typical size $N$. We determine the correct window $(u_N)_{N\geq 1}$ of the intensity parameter for which the capacity $\mathrm{cap}(\mathscr{I}(u_N)\cap D_N)$ starts to become negligible compared to $\mathrm{cap}(D_N)$; this roughly means that a random walk starting from far away starts to see through $\mathscr{I}(u_N)\cap D_N$. In the same spirit, we investigate the capacity of the simple random walk conditioned to stay in a large Euclidean ball up to time $t_N$, and find similar asymptotics by taking $t_N = u_N N^d$.

How thin does random interlacement have to be so that a random walk can see through it?

TL;DR

The paper identifies a sharp threshold for when holes in the random interlacement trace become visible to a distant random walk, quantified by the window and the scale . It proves that the capacity ratio converges to or according as tends to or , with a nontrivial regime when the limit is finite; the threshold depends on dimension via . It then derives an analogous dichotomy for a simple random walk confined to a large ball up to time , showing that the ratio of the walk’s range capacity to the ball’s capacity vanishes if and tends to 1 if , with dimension-dependent refinements. The proofs hinge on capacity decompositions, harmonic/equilibrium measures, and a key proposition controlling the capacity of the random-walk range, complemented by spectral-analytic arguments for the confined-walk setting and a dimension-3 excursion-based approach. Overall, the results illuminate how thinning random obstacles interact with random walks and reveal a concrete, dimension-sensitive phase transition in the visibility of holes through random interlacements.

Abstract

The random interlacements at level has been introduced by Sznitman, as a Poissonian collection of independent simple random walk trajectories on , , with intensity . Since then, several works investigated the properties of the random interlacements intersected with large sets of~. In this paper, we study the asymptotic behavior of the capacity of , where is the blow up of a compact set , with typical size . We determine the correct window of the intensity parameter for which the capacity starts to become negligible compared to ; this roughly means that a random walk starting from far away starts to see through . In the same spirit, we investigate the capacity of the simple random walk conditioned to stay in a large Euclidean ball up to time , and find similar asymptotics by taking .

Paper Structure

This paper contains 16 sections, 19 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Our main question
  3. A motivation from the folding of random walks
  4. Some other related works
  5. Random interlacements: Proof of Theorem \ref{['th:ratio-cap-entrelac']}
  6. Preliminaries on capacities
  7. Main argument and key proposition
  8. Estimates on : proof of Proposition \ref{['lem:deviation-cap-infty']}
  9. Probability that
  10. Probability that is small
  11. Constrained random walk: Proof of Theorem \ref{['th:ratio-cap-RW-boule']}
  12. Some preliminaries on confining probabilities
  13. Upper bound on the ratio
  14. Lower bound for Theorem \ref{['th:ratio-cap-RW-boule']}
  15. Dimension four and higher
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $d \geq 3$ and let $(u_N)_{N\geq 1}$ be a sequence of positive numbers. For any compact regular set $D \subset \mathbb{R}^d$, we have the following convergences in $\mathbb{P}$-probability (and in $L^1$): Moreover, if $0<\liminf_{N\to\infty} u_N \Theta_N\leq \limsup_{N\to\infty} u_N \Theta_N <+\infty$, then

Theorems & Definitions (37)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Proposition 1.8
  • Proposition 2.1
  • Theorem 2.2
  • ...and 27 more