Diffusion for the periodic wind-tree model

Abstract

The periodic wind-tree model is an infinite billiard in the plane with identical rectangular scatterers placed at each integer point. We prove that independtly of the size of scatters and generically with respect to the angle, the polynomial diffusion rate in this billiard is 2/3. Résumé Diffusion du vent dans les arbresLe vent dans les arbres périodique est un billard infini construit de la manière suivante. On considère le plan dans lequel sont placés des obstacles rectangulaires identiques à chaque point entier. Une particule (identifiée à un point) se déplace en ligne droite (le vent) et rebondit de manière élastique sur les obstacles (les arbres). Nous prouvons qu'indépendamment de la taille des obstacles et génériquement par rapport à l'angle initial de la particule le coefficient de diffusion polynomial des orbites de ce billard est 2/3.

Nonvarying sums of Lyapunov exponents of Abelian differentials in low genus

“…For example, it is shown by Delecroix, Hubert and Lelièvre in [13] that Lyapunov exponents are responsible for the rate of diffusion in the wind-tree model, where the parameters of the obstacle correspond to picking a flat surface in a fixed stratum. One would like to know this escape rate not only for a specific choice of parameters nor for the generic value of parameters but for all parameters.…”

Lyapunov exponents of the Hodge bundle over strata of quadratic differentials with large number of poles

“…This theorem followed an insightful observation of [Kon97] that this sum is related to the degree of a Hodge subbundle, which was proven later in [For02]. This was a starting point to evaluate Lyapunov exponents in certain particularly symmetric cases, for example for square-tiled cyclic covers [FMZ14], [EKZ11] and triangle groups [BM10]; and to compute explicitly diffusion rate of wind-tree models [DHL14], [DZ15].…”

Recurrence and non‐ergodicity in generalized wind‐tree models

“…The following two results are closely related to Theorem 2 in [7] and Theorem 4.2 and Lemma 6.3 in [16]. For the completeness of exposition we include their proofs in Appendix A.…”

“…If then the corresponding billiard table we denote by . The recurrence, ergodicity and diffusion times of standard Ehrenfest wind‐tree model, with , were discussed recently in . In particular, it was recently shown that for every pair of parameters and almost every direction θ the billiard flow on is recurrent and non‐ergodic and its rate of diffusion is .…”