Explicit sufficient invariants for an interacting particle system

Abstract

We introduce a new class of interacting particle systems on a graphG. Suppose initially there areNi(0) particles at each vertexiofG, and that the particles interact to form a Markov chain: at each instant two particles are chosen at random, and if these are atadjacentvertices ofG, one particle jumps to the other particle's vertex, each with probability 1/2. The processNenters a death state after a finite time when all the particles are in someindependentsubset of the vertices ofG, i.e. a set of vertices with no edges between any two of them. The problem is to find the distribution of the death state, ηi=Ni(∞), as a function ofNi(0).We are able to obtain, for some special graphs, thelimitingdistribution ofNiif the total number of particlesN→ ∞ in such a way that the fraction,Ni(0)/S= ξi, at each vertex is held fixed asN→ ∞. In particular we can obtain the limit law for the graphS2, the two-leaf star which has three vertices and two edges.