A confined random walk locally looks like tilted random interlacements (preprint)

Abstract

In this paper we consider the simple random walk on $\ZZ^d$, $d \geq 3$, conditioned to stay 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 Teixeira and Li & Sznitmann defined as tilted random interlacements. This tilted interlacement can be described as random interlacements but with trajectories given by random walks on conductances $c_N(x,y) = \phi_N(x) \phi_N(y)$, 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 in the work of Popov & Teixeira and used by Cerny & Teixeira to prove the well-known coupling between the simple random walk on the torus and the random interlacements.