On the nature of the Swiss cheese in dimension $3$

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 [BBH01], including their sharp estimates of the rate functions in the neighborhood of the origin.Remarkably the Swiss cheese heuristic also highlight a crucial difference between dimension 3 and dimensions 5 and higher. Indeed, in dimension three the typical scenario is time homogeneous, in the sense that the Wiener sausage considered up to time t, would spend all its time localized in a region of typical scale (t/ε) 1/3 , filling a fraction of order ε of every volume element, when the