Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-27T04:23:30.044Z Has data issue: false hasContentIssue false
  • English
  • Français

Explicit sufficient invariants for an interacting particle system

Part of: Special processes Graph theory Markov processes

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: itoh@ism.ac.jp
∗∗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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arnold, V. I. (1989). Mathematical Models of Classical Mechanics. Springer, New York.Google Scholar
Bogoyavlensky, O. I. (1988). Integrable discretization of the KdV equation. Phys. Lett. A 134, 34–38.Google Scholar
Bramson, M., and Griffeath, D. (1989). Flux and fixation in cyclic particle systems. Ann. Prob. 17, 26–45.Google Scholar
Ethier, S. N. (1976). A class of degenerate diffusion processes occurring in population genetics. Comm. Pure. Appl. Math. 29, 483–493.Google Scholar
Ethier, S. N. Kurtz, T. G. (1986). Markov Processes. Wiley, New York.Google Scholar
Flaschka, H. (1974). The Toda lattice I. Phys. Rev. B 9, 1924–1925.Google Scholar
Henon, M. (1974). Integrals of the Toda lattice. Phys. Rev. B 9, 1921–1923.Google Scholar
Itoh, Y. (1979). Random collision models in oriented graphs. J. Appl. Prob. 16, 36–44.CrossRefGoogle Scholar
Itoh, Y. (1993). Stochastic model of an integrable nonlinear system. J. Phys. Soc. Japan 62, 1826–1828.Google Scholar
Lax, P. D. (1968). Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math. 21, 467–490.Google Scholar
Liggett, T. M. (1985). Interacting Particle Systems. Springer, New York.Google Scholar
McKean, H. P. (1969). Stochastic Integrals. Academic Press, New York.Google Scholar
Moran, P. A. P. (1958). Random processes in genetics. Proc. Camb. Phil. Soc. 54, 60–71.Google Scholar
Nei, M., Maruyama, T., and Wu, C. (1982). Models of evolution of reproductive isolation. Genetics 103, 557–579.Google Scholar
Sato, K. (1976). Diffusion processes and a class of Markov chains related to population genetics. Osaka J. Math. 13, 631–659.Google Scholar
Whittaker, E. T., and Watson, G. N. (1927). A Course of Modern Analysis, 4th edn. Cambridge University Press, Cambridge, Section 11.8.Google Scholar