1998/09/01 by Yoshiaki Itoh, Colin Mallows, Colin L. Mallows +2 · 1 citation
Mathematics · Physics and Astronomy · #Stochastic processes and statistical mechanics #Markov Chains and Monte Carlo Methods #Theoretical and Computational Physics
paper · doi:10.1239/jap/1032265211
We introduce a new class of interacting particle systems on a graph G . Suppose initially there are 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 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, η i = N i (∞), as a function of N i (0). We are able to obtain, for some special graphs, the limiting distribution of N i if the total number of particles N → ∞ in such a way that the fraction, N i (0)/ S = ξ i , at each vertex is held fixed as N → ∞. In particular we can obtain the limit law for the graph S 2 , the two-leaf star which has three vertices and two edges.