Empires and Percolation: Stochastic Merging of Adjacent Regions

TL;DR

This paper introduces a stochastic merging model for adjacent planar regions, exploring conditions for the emergence of infinite regions and providing analytical and simulation evidence for a phenomenon called hegemony.

Contribution

It proposes a new stochastic model for region merging and establishes conditions for the appearance of infinite regions, connecting to percolation theory.

Findings

Conditions for hegemony are identified.

Simulations support the occurrence of hegemony.

Analytic arguments suggest hegemony in the uniform case.

Abstract

We introduce a stochastic model in which adjacent planar regions $A, B$ merge stochastically at some rate $\lambda(A,B)$, and observe analogies with the well-studied topics of mean-field coagulation and of bond percolation. Do infinite regions appear in finite time? We give a simple condition on $\lambda$ for this {\em hegemony} property to hold, and another simple condition for it to not hold, but there is a large gap between these conditions, which includes the case $\lambda(A,B) \equiv 1$. For this case, a non-rigorous analytic argument and simulations suggest hegemony.