Two-dimensional random interlacements: 0-1 law and the vacant set at criticality

TL;DR

This paper refines the proof that the vacant set of 2D random interlacements is infinite at criticality and establishes a zero-one law for tail events, advancing understanding of the model's phase transition.

Contribution

It corrects and simplifies a previous proof about the infinite nature of the vacant set at criticality and introduces a zero-one law for tail events in the model.

Findings

The vacant set at criticality is infinite.

A zero-one law applies to certain tail events.

The proof of the critical behavior is streamlined and corrected.

Abstract

We correct and streamline the proof of the fact that, at the critical point $\alpha=1$, the vacant set of the two-dimensional random interlacements is infinite (Comets, Popov, 2017). Also, we prove a zero-one law for a natural class of tail events related to the random interlacements.