Random interlacements on transient weighted graphs: 0-1 laws and FKG inequality

TL;DR

This paper investigates properties of the random interlacement model on transient weighted graphs, establishing the FKG inequality and 0-1 laws for non-local events, enhancing understanding of probabilistic behaviors in such structures.

Contribution

It provides a simple proof of the FKG property and demonstrates 0-1 laws for non-local events without additional assumptions.

Findings

Established the FKG inequality for the model

Proved 0-1 laws for certain non-local events

Extended understanding of probabilistic properties on weighted graphs

Abstract

We study some properties of the random interlacement model on a transient weighted graph, which was introduced by A. Teixeira in ["Interlacement percolation on transient weighted graphs", Augusto Teixeira, Electronic Journal of Probability (2009)]. We give a simple proof of the FKG-property and discuss the occurrence of several 0-1 laws for non-local events. We show in particular a 0-1 law for some increasing non-local events, without any assumption.