Percolation of terraces, and enhancements for the orthant model

TL;DR

This paper investigates a percolation model in high dimensions with a focus on how the critical probability varies with dimension, utilizing enhancement techniques and studying terrace geometries to establish monotonicity.

Contribution

It introduces a novel application of enhancement methods to a multi-environment percolation model and proves the strict monotonicity of the critical parameter with respect to dimension.

Findings

Proves $p_c(d)$ is strictly increasing with dimension $d$.

Develops a new framework for analyzing terraces in higher-dimensional percolation.

Extends percolation theory to models with multiple environments and phase transitions.

Abstract

We study a model of an i.i.d.~random environment in general dimensions $d\ge 2$, where each site is equipped with one of two environments. The model comes with a parameter $p$ which governs the frequency of the first environment, and for each dimension $d$ there is a critical parameter $p_c(d)$ at which there is a phase transition for the geometry of a particular connected cluster (the cluster is infinite for all $p$). We use the celebrated methodology of enhancements in this novel setting to prove that $p_c(d)$ is strictly monotone in $d$ for this model. To do so we study the discrete geometry and percolation theory of higher-dimensional structures called terraces.