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

Françoise Pene (1)

1 LM - Laboratoire de mathématiques de Brest

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).

Mots clés: Sinai billiard; Lorentz process; self intersection; variance

Domaines: Systèmes dynamiques [math.DS]; Probabilités [math.PR]

Françoise Pene. An asymptotic estimate of the variance of the self-intersections of a planar periodic Lorentz process. 2013. ⟨hal-00799504⟩