The vacant set of two-dimensional critical random interlacement is infinite

TL;DR

This paper proves that in the critical two-dimensional random interlacement model, the vacant set is almost surely infinite, and establishes regularity properties of the entrance measure for simple random walk on annular domains.

Contribution

It demonstrates the infiniteness of the vacant set at criticality and analyzes entrance measure regularity, addressing an open problem and providing new insights.

Findings

Vacant set is almost surely infinite at criticality

Entrance measure exhibits regularity properties

Addresses an open problem from prior research

Abstract

For the model of two-dimensional random interlacements in the critical regime (i.e., $\alpha=1$), we prove that the vacant set is a.s.\ infinite, thus solving an open problem from arXiv:1502.03470. Also, we prove that the entrance measure of simple random walk on annular domains has certain regularity properties; this result is useful when dealing with soft local times for excursion processes.