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

Černý, Jiří and Hayder, Thomas. (2021) Critical window for the vacant set left by random walk on the configuration model. Preprints Fachbereich Mathematik, 2021 (16).

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Official URL: https://edoc.unibas.ch/84117/

We study the simple random walk on the configuration model with given degree sequence $(d_1^n, \dots ,d_n^n)$ 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 Erdős-Rényi random graph.