Self-intersections of trajectories of Lorentz process

TL;DR

This paper investigates the long-term behavior of self-intersections in the trajectories of a periodic Lorentz process with convex obstacles, providing detailed statistical estimates and convergence results.

Contribution

It offers new precise estimates for the expectation and variance of self-intersections and proves their almost sure convergence under normalization.

Findings

Explicit estimates for expectation and variance of self-intersections

Almost sure convergence of normalized self-intersections

Enhanced understanding of Lorentz process trajectory complexity

Abstract

We study the asymptotic behaviour of the number of self-intersections of a trajectory of a periodic planar Lorentz process with strictly convex obstacles and finite horizon. We give precise estimates for its expectation and its variance. As a consequence, we establish the almost sure convergence of the self-intersections with a suitable normalization.