 
Summary: IOP PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND THEORETICAL
J. Phys. A: Math. Theor. 43 (2010) 025001 (10pp) doi:10.1088/17518113/43/2/025001
Empires and percolation: stochastic merging of
adjacent regions
D J Aldous1,2
, J R Ong and W Zhou
Department of Statistics, 367 Evans Hall # 3860, U.C. Berkeley, CA 94720, USA
Email: aldous@stat.berkeley.edu
Received 11 October 2009, in final form 5 November 2009
Published 10 December 2009
Online at stacks.iop.org/JPhysA/43/025001
Abstract
We introduce a stochastic model in which adjacent planar regions A, B merge
stochastically at some rate (A, B) and observe analogies with the wellstudied
topics of meanfield coagulation and of bond percolation. Do infinite regions
appear in finite time? We give a simple condition on for this 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 (A, B) 1. For
this case, a nonrigorous analytic argument and simulations suggest hegemony.
PACS number: 64.60.ah
