We introduce a new class of interacting particle systems on a graph G. Suppose there are initially N_{i} (0) particles at each vertex i of G, and that the particles interact to form a Markov chain: at each instant two particles are chosen at random, and if these are at adjacent vertices of G, one particle jumps to the other particle's vertex, each with probability 1/2. The process enters a death state after a finite time when all the particles are in some independent subset of the vertices of G, i.e., a set of vertices with no edge between any two of them. The problem is to find the distribution of the death state h _{i}= N_{i} (¥) as a function of the numbers N_{i} (0).
We are able to obtain, for some special graphs, the limiting distribution of each N_{i} if the total number of particles N ® ¥ in such a way that the fraction N_{i} (0) /N = x_{i} at each vertex is held fixed as N ® ¥. In particular we can obtain the limit law for the graph S_{2}: ··· having 3 vertices and 2 edges.
This talk is based on a joint paper with Colin Mallows and Larry Shepp [1]
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This document was translated from L^{A}T_{E}X by H^{E}V^{E}A.