Explicit sufficient invariants for an interacting particle system

We introduce a new class of interacting particle systems on a graph G. Suppose initially there are Ni (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 N 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 edges between any two of them. The problem is to find the distribution of the death state, ηi = Ni (∞), as a function of Ni (0). We are able to obtain, for some special graphs, the limiting distribution of Ni if the total number of particles N →∞ in such a way that the fraction, Ni (0)/S = ξi , at each vertex is held fixed as N →∞ . In particular we can obtain the limit law for the graph S2, the two-leaf star which has three vertices and two edges.