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Recent Results for the Stepping Stone Model

  • J. Theodore Cox
  • David Griffeath
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 8)

Abstract

Let G be a graph, and C = {0,1,...,κ-1} a set of colors. We want to discuss a continuous time random process ζt with state space CG = configurations of colors from C on the graph G. This system ζt, known as the stepping stone model, has very simple dynamics: in any state ζ and at any site x, the color at that site waits a mean 1/2 exponential holding time and then paints a randomly chosen neighboring vertex with its color. We will also be treating a companion process ξt on {subsets of G} called coalescing random walks. As the name implies, ξt consists of rate 1/2 continuous time random walks on G which coalesce when they collide. Let us write ξ t A to denote the evolution of those walks which start on a subset A G. The graphs of principal interest to us are:
  • GN = the complete graph on {0,1,...,N-1},

  • Z N d = the d-dimensional integers with period N,

  • Zd = the d-dimensional integers.

Tropical interpretations of the dimension d are good therapy for a Minnesota February: Baja California (d=1), Society Islands (d=2), Caribbean Marine World (d=3), and of course Mathematical Physics (d > 4). The cardinality κ of the color set C might be 2 or 32,768 or even ∞.

Keywords

Complete Graph Occupation Time Voter Model Interact Particle System Society Island 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York, Inc. 1987

Authors and Affiliations

  • J. Theodore Cox
    • 1
  • David Griffeath
    • 2
  1. 1.Dept. of MathematicsSyracuse UniversitySyracuseUSA
  2. 2.Dept. of MathematicsUniversity of WisconsinMadisonUSA

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