Uniqueness and tube property for the Swiss cheese large deviations

TL;DR

This paper investigates the large deviations of a 3D simple random walk conditioned to visit fewer sites, proving the uniqueness of the rate function's minimizer and convergence of the empirical measure of the skeleton to this minimizer.

Contribution

It establishes the uniqueness of the minimizer of the rate function for certain deviations and shows convergence of the empirical measure of the skeleton under the conditioned law.

Findings

The rate function has a unique minimizer over probability measures modulo shifts.

The empirical measure of the skeleton converges to this minimizer under the conditioned law.

The results apply to deviations of the range well below the mean.

Abstract

We consider the simple random walk on the Euclidean lattice, in three dimensions and higher, conditioned to visit fewer sites than expected, when the deviation from the mean scales like the mean. The associated large deviation principle was first derived in 2001 by van den Berg, Bolthausen and den Hollander in the continuous setting, that is for the volume of a Wiener sausage, and later taken up by Phetpradap in the discrete setting. One of the key ideas in their work is to condition the range of the random walk to a certain skeleton, that is a sub-sequence of the random walk path taken along an appropriate mesoscopic scale. In this paper we prove that (i) the rate function obtained by van den Berg, Bolthausen and den Hollander has a unique minimizer over the set of probability measures modulo shifts, at least for deviations of the range well below the mean, and (ii) the empirical measure of the skeleton converges under the conditioned law, in a certain manner, to this minimizer. To this end we use an adaptation of the topology recently introduced by Mukherjee and Varadhan to compactify the space of probability measures.