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Covering of an inner subset by the confined random walk

Nicolas Bouchot

TL;DR

This work analyzes the cover time of inner subsets by a simple random walk conditioned to stay in a large discrete domain, realized as a Doob transform using the first eigenfunction. By coupling the confined walk to tilted random interlacements, the authors derive a first-order covering-time asymptotics $\mathfrak{C}(\Lambda_N)\sim g(0)\alpha_\Lambda^{-1} \log|\Lambda_N|$ and, under a strong regularity/level-set deformation assumption, establish Gumbel fluctuations with a Poisson point process describing late points on the boundary level set. A detailed toolbox is developed for the tilted RI setting, including capacity, Green function, and decoupling bounds, which then carries over to the confined walk via a coupling theorem that ties the confined-range to RI traces within large balls. The results extend the torus-covering theory to inhomogeneous, drifted domains, revealing a nontrivial log-log correction and a boundary-driven Poissonization of late points, with explicit formulas for the associated intensity measures. Special cases such as balls and 1D segments illustrate the geometric impact of the level-set structure on cover times and late-point distributions.

Abstract

We consider the simple random walk conditioned to stay forever in a finite domain $D_N \subset \mathbb{Z}^d, d \geq 3$ of typical size $N$. This confined walk is a random walk on the conductances given by the first eigenvector of the Laplacian on $D_N$. On inner sets of $D_N$, the trace of this confined walk can be approximated by tilted random interlacements, which is a useful tool to understand some properties of the walk. In this paper, we propose to study the cover time of inner subsets $Λ_N$ of $D_N$ as well as the so-called late points of these subsets. If $Λ_N$ contains enough late points, we obtain the asymptotic expansion of the covering time as $c_ΛN^d \big[ \log N - \log\log N + \mathcal{G} \big]$, with $\mathcal{G}$ a Gumbel random variable, as well as a Poisson repartition of these late points. The method we use is similar to Belius' work about the simple random walk on the torus, which displays the same asymptotics albeit without the $\log \log N$ term. In the more general setting of ``ball-like'' $Λ_N$, we simply get the first term of the asymptotic expansion.

Covering of an inner subset by the confined random walk

TL;DR

This work analyzes the cover time of inner subsets by a simple random walk conditioned to stay in a large discrete domain, realized as a Doob transform using the first eigenfunction. By coupling the confined walk to tilted random interlacements, the authors derive a first-order covering-time asymptotics and, under a strong regularity/level-set deformation assumption, establish Gumbel fluctuations with a Poisson point process describing late points on the boundary level set. A detailed toolbox is developed for the tilted RI setting, including capacity, Green function, and decoupling bounds, which then carries over to the confined walk via a coupling theorem that ties the confined-range to RI traces within large balls. The results extend the torus-covering theory to inhomogeneous, drifted domains, revealing a nontrivial log-log correction and a boundary-driven Poissonization of late points, with explicit formulas for the associated intensity measures. Special cases such as balls and 1D segments illustrate the geometric impact of the level-set structure on cover times and late-point distributions.

Abstract

We consider the simple random walk conditioned to stay forever in a finite domain of typical size . This confined walk is a random walk on the conductances given by the first eigenvector of the Laplacian on . On inner sets of , the trace of this confined walk can be approximated by tilted random interlacements, which is a useful tool to understand some properties of the walk. In this paper, we propose to study the cover time of inner subsets of as well as the so-called late points of these subsets. If contains enough late points, we obtain the asymptotic expansion of the covering time as , with a Gumbel random variable, as well as a Poisson repartition of these late points. The method we use is similar to Belius' work about the simple random walk on the torus, which displays the same asymptotics albeit without the term. In the more general setting of ``ball-like'' , we simply get the first term of the asymptotic expansion.

Paper Structure

This paper contains 33 sections, 28 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Motivations
  3. Random walk confined in a large domain as a Doob transform
  4. Covering of inner sets - main result
  5. General setting
  6. Gumbel and Poisson behaviour under a stronger assumption
  7. Comparison with the random walk on the torus
  8. Acknowledgements
  9. Tilted interlacements, capacity and first estimates
  10. Tilting of random interlacements
  11. Cover level of the tilted interlacements
  12. Some useful estimates
  13. Useful facts about the confined walk and eigenfunction
  14. Probabilistic interpretation as a confined walk
  15. Properties of the eigenvector
  16. ...and 18 more sections

Key Result

Theorem 1.1

Under $\mathbf{P}^N_{\phi_N^2}$, we have the following asymptotics in probability: Moreover, $\mathfrak{C}_N(\Lambda_N)$ has "super-Gumbel" fluctuation, in the sense that for any fixed $z \in \mathbb{R}$,

Theorems & Definitions (50)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: Cover level of the interlacements
  • Theorem 2.2
  • Proposition 2.3
  • Corollary 2.4
  • Proposition 3.1: Regularity
  • Proposition 3.2: Convergence
  • Proposition 3.3: Last exit decomposition
  • proof
  • ...and 40 more