Phase transitions for degenerate random environments

TL;DR

This paper investigates phase transitions in a class of i.i.d. random environment models across multiple dimensions, demonstrating sharp changes in cluster connectivity as a parameter varies, even in non-monotone cases.

Contribution

It establishes the existence of sharp phase transitions in the geometry of clusters for a broad class of random environment models, including non-monotone cases.

Findings

Models exhibit sharp phase transitions in cluster connectivity.

Phase transition occurs as the parameter p varies.

Includes analysis of non-monotone environment models.

Abstract

We study a class of models of i.i.d.~random environments in general dimensions $d\ge 2$, where each site is equipped randomly with an environment, and a parameter $p$ governs the frequency of certain environments that can act as a barrier. We show that many of these models (including some which are non-monotone in $p$) exhibit a sharp phase transition for the geometry of connected clusters as $p$ varies.