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Explicit sufficient invariants for an interacting particle system

Part of: Special processes Markov processes Graph theory

Published online by Cambridge University Press:  14 July 2016

Yoshiaki Itoh ,
Colin Mallows  and
Larry Shepp
Yoshiaki Itoh*
Affiliation:
Institute of Statistical Mathematics, Tokyo
Colin Mallows*
Affiliation:
AT&T Laboratories, Florham Park
Larry Shepp*
Affiliation:
Rutgers University
*
∗Postal address: The Institute of Statistical Mathematics, 4–6–7 Minami-Azabu Minato-ku, Tokyo 106–8589, Japan. Email address: [email protected]
∗∗Postal address: AT&T Laboratories, 180 Park Avenue, Florham Park, NJ 07932, USA.
∗∗∗Postal address: Department of Statistics, Rutgers University, Hill Center, Piscataway, NJ 08903, USA.
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Abstract

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.

Keywords

Interacting particle systemsstochastic invariantsgraph processes

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 05C80: Random graphs
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 3 , September 1998 , pp. 633 - 641
Copyright
Copyright © Applied Probability Trust 1998 

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