Painting a graph with competing random walks

• Autores: Jason Miller
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 41, Nº. 2, 2013, págs. 636-670
• Idioma: inglés
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• Resumen

  • Let X1, X2 be independent random walks on Zdn, d≥3, each starting from the uniform distribution. Initially, each site of Zdn is unmarked, and, whenever Xi visits such a site, it is set irreversibly to i. The mean of |Ai|, the cardinality of the set Ai of sites painted by i, once all of Zdn has been visited, is 12nd by symmetry. We prove the following conjecture due to Pemantle and Peres: for each d≥3 there exists a constant αd such that limn→∞Var(|Ai|)/hd(n)=14αd where h3(n)=n4, h4(n)=n4(logn) and hd(n)=nd for d≥5. We will also identify αd explicitly and show that αd→1 as d→∞. This is a special case of a more general theorem which gives the asymptotics of Var(|Ai|) for a large class of transient, vertex transitive graphs; other examples include the hypercube and the Caley graph of the symmetric group generated by transpositions.