A confined random walk locally looks like tilted random interlacements

TL;DR

This paper demonstrates that a confined simple random walk in a large domain behaves like a tilted random interlacement, with the tilt determined by the domain's principal eigenvector, over sufficiently long time scales.

Contribution

It establishes a coupling between confined random walks and tilted random interlacements using soft local times methodology, extending previous coupling results.

Findings

Coupling holds for time scales $t_N extgreater N^{2+eta}$.

Tilted interlacements are characterized by conductances based on the domain's eigenvector.

The approach generalizes previous couplings to confined domains with eigenvector-based tilts.

Abstract

In this paper we consider the simple random walk on $\mathbb{Z}^d$, $d \geq 3$, 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 (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 $c_N(x,y) = \phi_N(x) \phi_N(y)$, 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 (2015) and used by \v{C}ern\'y & Teixeira (2016) to prove the well-known coupling between the simple random walk on the torus and the random interlacements.