Logarithmic components of the vacant set for random walk on a discrete torus.

Windisch, David

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We study the trajectory of a simple random walk on a -regular graph with ≥ 3 and locally tree-like structure as the number of vertices grows. Examples of such graphs include random -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 , where > 0 is a fixed positive parameter. We show that this so-called set exhibits a phase transition in in the following sense: there exists...

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