Critical window for the vacant set left by random walk on the configuration model

We study the simple random walk on the configuration model with given degree sequence (dn1 , ... , dnn) and investigate the connected components of its vacant set at level u > 0. We show that the size of the maximal connected component exhibits a phase transition at level u* which can be related with the critical parameter of random interlacements on a certain Galton-Watson tree. We further show that there is a critical window of size n-1/3 around u* in which the largest connected components of the vacant set have a metric space scaling limit resembling the one of the critical Erdos-Renyi random graph.