Critical window for the vacant set left by random walk on random regular graphs

Details Critical window for the vacant set left by random walk on random regular graphs Critical window for the vacant set left by random walk on random regular graphs

2011-01-10

arxiv.org/abs/1101.1978v1

Mathematics

Probability

22 pages, 1 figure

Scientific paper

We consider the simple random walk on a random d-regular graph with n vertices, and investigate percolative properties of the set of vertices not visited by the walk until time un, where u > 0 is a fixed positive parameter. It was shown in [arXiv:1012.5117] that this so-called vacant set exhibits a phase transition at u = u*: there is a giant component if u < u* and only small components when u > u*. In this paper we show the existence of a critical window of size n^(-1/3) around u*. In this window the size of the largest cluster is of order n^(2/3).

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