Random walks in confined geometry: convex hull and cover times
Marie Chupeau (LPTMC, UPMC, Paris)
What is the impact of a confinement on random walks? To answer this question, I will first focus on a partial confinement, an infinite reflecting wall, and study how the space taken by a planar random walk is affected by the presence of the wall. The space explored by a random walk in the plane can be characterized by its convex hull, defined as the minimum convex polygon enclosing the trajectory, and in particular by its perimeter. After having studied a one-dimensional version of the problem, I will show that the mean perimeter of the convex hull of a planar random walk exhibits a non-trivial dependance on the initial distance of the walker to the wall, in sharp contrast with the results in one dimension.
Second, I will consider a random walk in total confinement and study the time it takes to visit all sites of a given domain. This time, known as the cover time, is a key observable to quantify the efficiency of exhaustive searches, which require a complete exploration of an area and not only the discovery of a single target. Despite its broad relevance, in biology, ecology, or robotics, the cover time has remained elusive and so far, explicit results have been scarce and mostly limited to regular random walks. I will show how to determine the distribution of the cover time for a broad range of random search processes, and highlight its universal features. I will conclude with an application of this result and discuss how the cover time can be optimized for some research strategies.