Moderate deviations for the range of a transient random walk: path concentration

TL;DR

This paper investigates how a transient random walk on a lattice can deviate downward to minimize its range boundary, revealing the optimal strategies and path behaviors involved.

Contribution

It introduces new techniques to analyze the boundary deviations of the walk's range and applies them to describe the pathwise structure of the boundary shrinking.

Findings

Identifies the optimal strategy for boundary reduction.

Provides pathwise descriptions of the boundary shrinking behavior.

Develops techniques applicable to the entire range, not just the boundary.

Abstract

We study downward deviations of the boundary of the range of a transient walk on the Euclidean lattice. We describe the optimal strategy adopted by the walk in order to shrink the boundary of its range. The technics we develop apply equally well to the range, and provide pathwise statements for the {\it Swiss cheese} picture of Bolthausen, van den Berg and den Hollander \cite{BBH}.