A confined random walk locally looks like tilted random interlacements

Abstract

In this paper we consider the simple random walk on ${\mathbb{Z}}^d$, $d\ge 3$, conditioned to stay in a large domain $D_N$ of typical diameter $N$. Considering the range up to time $t_N\ge N^{2+\hat{I}´}$ for some $\hat{I}´>0$, we establish a coupling with what Teixeira (2009) and Li & Sznitman (2014) defined as "tilted random interlacements". This tilted interlacement can be described as random interlacements but with trajectories given by random walks on conductances $c_N(x,y)=\ddot{I}{•}_N(x)\ddot{I}{•}_N(y)$, where $\ddot{I}{•}_N$ is the first eigenvector of the discrete Laplace-Beltrami operator on $D_N$. The coupling follows the methodology of the soft local times, introduced by Popov & Teixeira (2015) and used by Černý & Teixeira (2016) to prove the well-known coupling between the simple random walk on the torus and the random interlacements.

44 pages, 2 figures ; v2: Corrected Lemma 3.2 (the invariant measures are not the same, but are close)