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arXiv · 1012.5117

Giant vacant component left by a random walk in a random d-regular graph

Jiri Cerny·Augusto Teixeira·David Windisch
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Abstract

We study the trajectory of a simple random walk on a d-regular graph with d>2 and locally tree-like structure as the number n of vertices grows. Examples of such graphs include random d-regular graphs and large girth expanders. For these graphs, we investigate percolative properties of the set of vertices not visited by the walk until time un, where u>0 is a fixed positive parameter. We show that this so-called vacant set exhibits a phase transition in u in the following sense: there exists an explicitly computable threshold u* such that, with high probability as n grows, if u u*, then it has a volume of order log(n). The critical value u* coincides with the critical intensity of a random interlacement process (introduced by Sznitman [arXiv:0704.2560]) on a d-regular tree. We also show that the random interlacement model describes the structure of the vacant set in local neighbourhoods.

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Jiri Cerny, Augusto Teixeira, David Windisch. 2010-12-22. Giant vacant component left by a random walk in a random d-regular graph. https://doi.org/10.1214/10-aihp407

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