  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).
