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We consider the simple random walk on a random d-regular graph with nvertices, and investigate percolative properties of the set of vertices notvisited 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 aphase transition at u = u*: there is a giant component if u < u* and only smallcomponents when u > u*. In this paper we show the existence of a criticalwindow of size n^(-1/3) around u*. In this window the size of the largestcluster is of order n^(2/3).

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