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A confined random walk locally looks like tilted random interlacements

Nicolas Bouchot

TL;DR

The paper proves a local coupling between the range of a simple random walk conditioned to stay in a large domain and tilted random interlacements, by embedding the confined walk into a conductance-based Markov chain with transition kernel derived from the first Dirichlet eigenvector. It develops the soft local times coupling to compare entrance/exit sites and excursions, establishing high-probability inclusions between the confined-walk range and a tilted interlacement at an appropriate level $u_N$. The approach leverages precise eigenvector estimates, mixing-time bounds, and variance control of soft-local-time functionals to transfer interlacement geometry to the conditioned walk, with potential applications to disconnection and covering-type problems in the confined setting. Overall, the work extends the Teixeira-type couplings to a non-Markovian conditioning via a time-homogenized, inhomogeneous interlacement framework, providing quantitative locally-RI-like descriptions for the confined walk.

Abstract

In this paper we consider the simple random walk on $\mathbb{Z}^d$, $d \geq 3$, conditioned to stay in a large domain $D_N$ of typical diameter $N$. Considering the range up to time $t_N \geq N^{2+δ}$ for some $δ> 0$, we establish a coupling with what Teixeira (2009) and Li & Sznitman (2014) defined as "tilted random interlacements". This tilted interlacement can be described as random interlacements but with trajectories given by random walks on conductances $c_N(x,y) = φ_N(x) φ_N(y)$, where $φ_N$ is the first eigenvector of the discrete Laplace-Beltrami operator on $D_N$. The coupling follows the methodology of the soft local times, introduced by Popov & Teixeira (2015) and used by Černý & Teixeira (2016) to prove the well-known coupling between the simple random walk on the torus and the random interlacements.

A confined random walk locally looks like tilted random interlacements

TL;DR

The paper proves a local coupling between the range of a simple random walk conditioned to stay in a large domain and tilted random interlacements, by embedding the confined walk into a conductance-based Markov chain with transition kernel derived from the first Dirichlet eigenvector. It develops the soft local times coupling to compare entrance/exit sites and excursions, establishing high-probability inclusions between the confined-walk range and a tilted interlacement at an appropriate level . The approach leverages precise eigenvector estimates, mixing-time bounds, and variance control of soft-local-time functionals to transfer interlacement geometry to the conditioned walk, with potential applications to disconnection and covering-type problems in the confined setting. Overall, the work extends the Teixeira-type couplings to a non-Markovian conditioning via a time-homogenized, inhomogeneous interlacement framework, providing quantitative locally-RI-like descriptions for the confined walk.

Abstract

In this paper we consider the simple random walk on , , conditioned to stay in a large domain of typical diameter . Considering the range up to time for some , we establish a coupling with what Teixeira (2009) and Li & Sznitman (2014) defined as "tilted random interlacements". This tilted interlacement can be described as random interlacements but with trajectories given by random walks on conductances , where is the first eigenvector of the discrete Laplace-Beltrami operator on . The coupling follows the methodology of the soft local times, introduced by Popov & Teixeira (2015) and used by Černý & Teixeira (2016) to prove the well-known coupling between the simple random walk on the torus and the random interlacements.
Paper Structure (35 sections, 26 theorems, 262 equations, 2 figures)

This paper contains 35 sections, 26 theorems, 262 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Simple random walk on the torus and random interlacements
  3. Simple random walk conditioned to stay in a large domain, confined walk
  4. Motivations and first notations
  5. The main object of study: the confined random walk
  6. Some preliminaries on random interlacements
  7. Definition of random interlacements
  8. Tilting of random interlacements
  9. Main result
  10. Connection with other works
  11. Possible applications of the coupling
  12. Disconnection of sets
  13. Disconnection of cylinders
  14. Covering time
  15. Organization of the paper
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let $\delta \in (0,1)$ and consider a sequence $(t_N)_{N\geq 1}$ that satisfies $t_N/ N^{2+\delta} \to +\infty$. We define as well as $\varepsilon_N \vcentcolon= N^{-\delta/4} \to 0$. Then, there are some $\eta > 0$ and some constants $c_1,c_2>0$ (that only depend on $\alpha, \delta, \varepsilon, D$ and $d \geq 3$) and a coupling $\mathbb{Q}$ of $\mathcal{R}_{\phi_N}(t_N)$ and $\mathscr{I}_{\Psi_

Figures (2)

  • Figure 1: Representation of $D_N$ and $B_N$ with its $\varepsilon N$-neighborhood, and a random walk path conditioned to stay in $D_N$.
  • Figure 2: Some excursions of the confined RW (left) and of two trajectories $w^j, w^{j'}$ of the tilted RI (right).

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 2.1: teixeiracoupling
  • Theorem 2.2
  • proof
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 45 more