Traces left by random walks on large random graphs: local limits

Abstract: In this article, we develop a theory for understanding the traces left by a random walk in the vicinity of a randomly chosen reference vertex. The analysis is related to interlacements but goes beyond previous research by showing weak limit theorems for the vicinity around the reference vertex together with the path segments of the random walk in this vicinity. Roughly speaking, our limit theorem requires appropriate mixing properties for the random walk together with a stronger variant of Benjamini-Schramm convergence for the underlying graph model. If these assumptions are satisfied, the limiting object can be explicitly given in terms of the Benjamini-Schramm limit of the graph model.