Ballistic Annihilation
What is the structure of the set of the last few points visited by a random
walk on a graph? We show that on vertex-transitive graphs of bounded
degree, this set is decorrelated (it is close to a product measure in total
variation) if and only if a simple geometric condition on the diameter of
the graph holds. In this case, the cover time has universal fluctuations:
properly scaled, this time converges to a Gumbel distribution.
To prove this result we rely on recent progress in geometric group theory
(about quantitative versions of Gromov's Theorem for finite
vertex-transitive graphs) and we prove refined quantitative estimates
showing that the hitting time of a small set of vertices is typically
approximately an exponential random variable.
This talk is based on joint work with Nathanaël Berestycki and Lucas
Teyssier.