An asymptotic estimate of the variance of the self-intersections of a planar periodic Lorentz process

TL;DR

This paper provides an asymptotic estimate of the variance of the number of self-intersections in a planar periodic Lorentz process, showing it grows proportionally to n^2, extending results known for random walks.

Contribution

It establishes the asymptotic behavior of the variance of self-intersections in a Lorentz process, a novel result for this class of dynamical systems.

Findings

Variance of self-intersections is asymptotically proportional to n^2.

The result parallels known findings for simple planar random walks.

Provides a key statistical property of Lorentz processes with convex obstacles.

Abstract

We consider a periodic planar Lorentz process with strictly convex obstacles and finite horizon. This process describes the displacement of a particle moving in the plane with unit speed and with elastic reflection on the obstacles. We call number of self-intersections of this Lorentz process the number V(n) of couples of integers (k,m) smaller than n such that the particle hits a same obstacle both at the k-th and at the m-th collision times. The aim of this paper is to prove that the variance of V(n) is equivalent to cn^2 (such a result has recently been proved for simple planar random walks by Deligiannidis and Utev).