Summary
Two of the simplest interacting particle systems are the coalescing random walks and the voter model. We are interested here in the asymptotic density and growth of these systems ast→∞. More specifically, let (ζ Zdt ) be a system of coalescing random walks with initial stateZ d, and (ζ Ot ) a voter model with a single individual originating atO. We analyse\(p_t = P(0 \in \zeta _t^{Zd} ) = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{P} (\zeta _t^0 \ne )\), and show that\(p_t \sim \frac{1}{\pi }\frac{{\log t}}{t}\) ast→∞ ford=2, andp t∼(γdt)−1 ast→∞ ford≧3 for someγ d. As a consequence, conditioned on non-extinction ofζ Ot , Pt¦ζ Ot ¦ approaches an exponential distribution. Results of a recent paper by Sawyer are applied.
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Bramson, M., Griffeath, D. Asymptotics for interacting particle systems onZ d . Z. Wahrscheinlichkeitstheorie verw Gebiete 53, 183–196 (1980). https://doi.org/10.1007/BF01013315
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DOI: https://doi.org/10.1007/BF01013315