On the nature of the Swiss cheese in dimension 3

TL;DR

This paper investigates the structure of the Swiss cheese phenomenon in three dimensions, analyzing how random walks meet or squeeze their range, revealing optimal meeting regions and strategies for range minimization.

Contribution

It introduces a new inequality for estimating the cost of visiting sparse sites and applies large deviation principles to analyze random walk behaviors in three dimensions.

Findings

Random walks most likely meet in regions of optimal density.

A single random walk tends to reach a small range through a time-uniform strategy.

The results rely on a novel inequality and sharp large deviation estimates.

Abstract

We study scenarii linked with the Swiss cheese picture in dimension three obtained when two random walks are forced to meet often, or when one random walk is forced to squeeze its range. In the case of two random walks, we show that they most likely meet in a region of optimal density. In the case of one random walk, we show that a small range is reached by a strategy uniform in time. Both results rely on an original inequality estimating the cost of visiting sparse sites, and in the case of one random walk on the precise Large Deviation Principle of van den Berg, Bolthausen and den Hollander, including their sharp estimates of the rate functions in the neighborhood of the origin