Covering of inner subsets by the confined random walk (Work in Progress)

Abstract

We consider the simple random walk that is conditioned to stay in a large domain (D_N = N \cdot D \cap \mathbb{Z}^d ) with typical size N, which is a Doob’s transform of simple random walk by the first eigenvector of the walk killed upon exiting D_N. Considering an inner subset A_N of D_N, we prove asymptotics for the covering time of A_N by the confined walk, meaning the time at which the walk has visited all the points of A_N. We get that the covering time is asymptotically N^d log N with a constant that depends on the minimal value of the eigenvector on A_N. We also prove that restricting to level sets of the eigenvector, we get Gumbel fluctuations and a convergence towards a Poisson point process. This work relies on a previous result from the same author which establishes a coupling between the confined walk and random interlacements, on which we can use the sketch of Belius’ proof for the covering time of SRW on the torus.