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

Migration Covariance 

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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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