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The critical value for some non-attractive long range nearest particle systems
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  • Published: March 1992

The critical value for some non-attractive long range nearest particle systems

  • T. S. Mountford1 

Probability Theory and Related Fields volume 93, pages 67–76 (1992)Cite this article

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Summary

We consider the nearest particle system which gives birth rate γ to each vacant interval, concentrated on the interval's midpoint(s). We prove that a critical value for γ exists and equals one. The proof extends to a large class of nearest particle systems. This paper solves a problem suggested by Liggett (1985).

In the following we deal with nearest particle systems {η t :t≦0}. These can be described as particle systems with the following flip rates:

$$\begin{gathered} for \eta (x) = 1, c(x,\eta ) = 1 \hfill \\ for \eta (x) = 0, c(x,\eta ) = \lambda \beta (l_x ,r_x ) \hfill \\ \end{gathered} $$

wherel x for a configuration η is the smallest positive 1 so that η(x-l)=1 andr x is similarly the smallest positiver so that η(x+r)=1. For a survey of the important properties and results see Liggett (1985), Chap. 7. In this paper we will be concerned with the question of survivability of infinite systems. In this paper we say the process survives ifP 1(η t (0)=1) is bounded away from O (here and throughout the paperP 1() will refer to probabilities starting with η(x)≡1).

Our ideas will apply to a wide range of nearest particle systems, but for concreteness we will consider the “midpoint” nearest particle system for which

$$\beta (l,r) = \begin{array}{*{20}c} {\lambda for l = r} \\ {{\lambda \mathord{\left/ {\vphantom {\lambda 2}} \right. \kern-\nulldelimiterspace} 2} for \left| {l - r} \right| = 1} \\ \end{array} $$

and β(l, r) equals zero in all other cases. For notational simplicity we will assume that we happen to only be dealing with intervals of even length so that the birth rate is always concentrated at a single point in an interval. This tidying assumption will not affect the validity of the proofs. We prove

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References

  • Bramson, M., Gray, L.: A note on the survival of the long-range contact process. Ann. Probab.9, 885–990 (1981).

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  • Bramson, M.: Survival of nearest particle systems with low birth rates. Ann. Probab.17, 433–444 (1989)

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  • Liggett, T.: Interacting particle systems. Berlin Heidelberg New York: Springer 1985

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  • Mountford, T.: The critical value for the uniform nearest particle system. Ann. Probab. (to appear)

  • Peyrière, J.: A singular random measure generated by splitting [0, 1]. Z. Wahrscheinlichkeitstheor. Verw. Geb.47, 289–297 (1979)

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, 90024, Los Angeles, CA, USA

    T. S. Mountford

Authors
  1. T. S. Mountford
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Research partially supported by NSF Grant DMS 91-57461

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Mountford, T.S. The critical value for some non-attractive long range nearest particle systems. Probab. Th. Rel. Fields 93, 67–76 (1992). https://doi.org/10.1007/BF01195388

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  • Received: 22 June 1990

  • Revised: 11 November 1990

  • Issue Date: March 1992

  • DOI: https://doi.org/10.1007/BF01195388

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Mathematics Subject Classification (1980)

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