One-dimensional random interlacements

TL;DR

This paper introduces a one-dimensional version of random interlacements using conditioned random walks, demonstrating convergence results and a central limit theorem for local times, extending the understanding of such stochastic processes.

Contribution

It defines the one-dimensional random interlacements and proves convergence of vacant sets and local times, providing new insights into their probabilistic structure.

Findings

Vacant set on the ring graph converges to the one-dimensional interlacements' vacant set.

A central limit theorem for the interlacements' local time is established.

Local times of the conditioned walk on the ring graph converge in law to the interlacements' local times.

Abstract

We base ourselves on the construction of the two-dimensional random interlacements [12] to define the one-dimensional version of the process. For this constructions we consider simple random walks conditioned on never hitting the origin, which makes them transient. We also compare this process to the conditional random walk on the ring graph. Our results are the convergence of the vacant set on the ring graph to the vacant set of one-dimensional random interlacements, a central limit theorem for the interlacements' local time for sites far from the origin and the convergence in law of the local times of the conditional walk on the ring graph to the interlacements' local times.