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

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)

On the class of caustics by reflection

Given any light position S in the complex projective plane P^2 and any algebraic curve C of P^2 (with any kind of singularities), we consider the incident lines coming from S (i.e. the lines containing S) and their reflected lines after reflection off the mirror curve C. The caustic by reflection is the Zariski clusure of the envelope of these reflected lines. We introduce the notion of reflected polar curve and express the class of the caustic by reflection in terms of intersection numbers of C with the reflected polar curve, thanks to a fundamental lemma established in [14]. This approach enables us to get an explicit formula for the class of the caustic by reflection in every case in terms of intersection numbers of the initial curve C.Comment: 21 pages, 1 figur

Random walk in random environment in a two-dimensional stratified medium with orientations

We consider a model of random walk in ${\mathbb Z}^2$ with (fixed or random) orientation of the horizontal lines (layers) and with non constant iid probability to stay on these lines. We prove the transience of the walk for any fixed orientations under general hypotheses. This contrasts with the model of Campanino and Petritis, in which probabilities to stay on these lines are all equal. We also establish a result of convergence in distribution for this walk with suitable normalizations under more precise assumptions. In particular, our model proves to be, in many cases, even more superdiffusive than the random walks introduced by Campanino and Petritis.Comment: 23 pages, 1 figur

Poisson law for some nonuniformly hyperbolic dynamical systems with polynomial rate of mixing

We consider some nonuniformly hyperbolic invertible dynamical systems which are modeled by a Gibbs-Markov-Young tower. We assume a polynomial tail for the inducing time and a polynomial control of hyperbolicity, as introduced by Alves, Pinheiro and Azevedo. These systems admit a physical measure with polynomial rate of mixing. In this paper we prove that the distribution of the number of visits to a ball B(x, r) converges to a Poisson distribution as the radius r $\to$ 0 and after suitable normalization.Comment: 21 pages, 3 figure

Degree and class of caustics by reflection for a generic source

We are interested in the study of caustics by reflection of irreducible algebraic planar curves (in the complex projective plane). We prove the birationality of the caustic map (for a generic light position). We also give simple formulas for the degree and the class of caustics by reflection valid for any irreducible algebraic curve of degree at least 2 and for a generic light position.Comment: 5 page

Recurrence rates and hitting-time distributions for random walks on the line

We consider random walks on the line given by a sequence of independent identically distributed jumps belonging to the strict domain of attraction of a stable distribution, and first determine the almost sure exponential divergence rate, as r goes to zero, of the return time to (-r,r). We then refine this result by establishing a limit theorem for the hitting-time distributions of (x-r,x+r) with arbitrary real x.Comment: Published in at http://dx.doi.org/10.1214/11-AOP698 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Convergence of U-statistics indexed by a random walk to stochastic integrals of a Levy sheet

We establish limit theorems for U-statistics indexed by a random walk on Z^d and we express the limit in terms of some Levy sheet Z(s,t). Under some hypotheses, we prove that the limit process is Z(t,t) if the random walk is transient or null-recurrent ant that it is some stochastic integral with respect to Z when the walk is positive recurrent. We compare our results with results for random walks in random scenery.Comment: 38 page