# Recent Results for the Stepping Stone Model

Chapter

## 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 ζ
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

_{t}with state space C^{G}= configurations of colors from C on the graph G. This system ζ_{t}, 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 ξ_{t}on {subsets of G} called coalescing random walks. As the name implies, ξ_{t}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:-
G

_{N}= the complete graph on {0,1,...,N-1}, -
Z

_{N}^{d}= the d-dimensional integers with period N, -
Z

^{d}= the d-dimensional integers.

*κ*of the color set C might be 2 or 32,768 or even ∞.### Keywords

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