Theoretical Statistics and Mathematics Unit, ISI Delhi
The path of a Brownian motion on a $d$-dimensional torus run up to time $t$ is a random compact subset of the torus. In this talk we look at the geometric and spectral properties of the complement $C(t)$ of this set when $t$ tends to infinity. Questions we address are the following:
1. What is the linear size of the largest region in $C(t)$?
2. What does $C(t)$ look like around this region?
3. Does $C(t)$ have some sort of "component-structure"?
4. What are the largest capacity, largest volume and smallest principal Dirichlet eigenvalue of the "components" of $C(t)$?
We discuss both $d \geq 3$ and $d=2$, which turn out to be very different.
Based on joint work with Michiel van den Berg (Bristol), Erwin Bolthausen (Zurich) and Jesse Goodman (Auckland).