Extremal geometry of a Brownian porous medium

TL;DR

This paper investigates the extremal geometric properties of the complement of a Brownian motion path on a high-dimensional torus, providing large deviation principles and identifying the shape and size of the largest regions.

Contribution

It introduces a detailed analysis of the extremal geometry of the Brownian path complement, including shape, size, and probabilistic behavior of large components, extending previous laws of large numbers.

Findings

Largest regions scale linearly with phi

Derived large deviation principles for inradius and cover time

Identified the shape and distribution of the largest components

Abstract

The path W[0,t] of a Brownian motion on a d-dimensional torus T^d run for time t is a random compact subset of T^d. We study the geometric properties of the complement T^d \ W[0,t] for t large and d >= 3. In particular, we show that the largest regions in this complement have a linear scale phi = [(d log t)/(d-2)kt]^{1/(d-2)}, where k is the capacity of the unit ball. More specifically, we identify the sets E for which T^d \ W[0,t] contains a translate of phi E, and we count the number of disjoint such translates. Furthermore, we derive large deviation principles for the largest inradius of T^d \ W[0,t] for t large and the epsilon-cover time of T^d for epsilon small. Our results, which generalise laws of large numbers proved by Dembo, Peres and Rosen, are based on a large deviation principle for the shape of the component with largest capacity in T^d \ W_rho[0,t], where W_rho[0,t] is the Wiener sausage of radius rho = rho(t), with rho(t) chosen much smaller than phi but not too small. The idea behind this choice is that T^d \ W[0,t] consists of "lakes", whose linear size is of order phi, connected by narrow "channels". We also derive large deviation principles for the principal Dirichlet eigenvalue and for the maximal volume of the components of T^d \ W_rho[0,t] for t large. Our results give a complete picture of the extremal geometry of T^d \ W[0,t] and of the optimal strategy for W[0,t] to realise the extremes.