A shape theorem for the orthant model

TL;DR

This paper proves a shape theorem for the orthant model, a random medium on  lattice, showing the growth shape of reachable sites when the probability of arrows pointing outward is high, using subadditivity methods.

Contribution

It establishes a shape theorem for the orthant model in general dimensions, overcoming the challenge of non-stationarity of the primary objects.

Findings

Shape theorem holds for large p in the orthant model.

Reachable set grows asymptotically to a deterministic shape.

Uses subadditivity despite non-stationarity of the process.

Abstract

We study a particular model of a random medium, called the orthant model, in general dimensions $d\ge 2$. Each site $x\in \Z^d$ independently has arrows pointing to its positive neighbours $x+e_i$, $i=1,\dots, d$ with probability $p$ and otherwise to its negative neighbours $x-e_i$, $i=1,\dots, d$ (with probability $1-p$). We prove a shape theorem for the set of sites reachable by following arrows, starting from the origin, when $p$ is large. The argument uses subadditivity, as would be expected from the shape theorems arising in the study of first passage percolation. The main difficulty to overcome is that the primary objects of study are not stationary, which is a key requirement of the subadditive ergodic theorem.