A confined random walk locally looks like tilted random interlacements

Overview & Synopsis

In this paper we consider the simple random walk on $\mathbb{Z}^d$, $d \geq 3$, conditioned to stay forever in a large domain $D_N$ of typical diameter $N$. Considering the range up to time $t_N \geq N^{2+\delta}$ for some $\delta > 0$, we establish a coupling with what Li \& Sznitman defined as \textit{tilted random interlacements}. This tilted interlacement can be described as random interlacements but with trajectories given by random walks on conductances $c_N(x,y) = \phi_N(x) \phi_N(y)$, where $\phi_N$ 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 and used by {\v C}ern{\'y} \& Teixeira to prove the well-known coupling between the simple random walk on the torus and the random interlacements.