Two-dimensional random interlacements and late points for random walks

TL;DR

This paper introduces a new two-dimensional random interlacements model based on conditioned random walks, linking it to late points in cover times and providing insights into the structure of the vacant set.

Contribution

It defines a novel 2D random interlacements model and connects it to late points in random walk cover times, offering a microscopic understanding.

Findings

The model of 2D random interlacements is well-defined and characterized.

The law of the vacant set around the origin approximates the interlacement at corresponding levels.

Provides a new perspective on the structure of late points in random walks.

Abstract

We define the model of two-dimensional random interlacements using simple random walk trajectories conditioned on never hitting the origin, and then obtain some properties of this model. Also, for random walk on a large torus conditioned on not hitting the origin up to some time proportional to the mean cover time, we show that the law of the vacant set around the origin is close to that of random interlacements at the corresponding level. Thus, this new model provides a way to understand the structure of the set of late points of the covering process from a microscopic point of view.