Nous considérons la région obtenue en enlevant de les disques de rayon , centrés aux points de coordonnées entières avec . Nous étudions la répartition de la longueur du libre parcours (temps de sortie) d’une particule ponctuelle, partant de sur une trajectoire rectiligne de direction quand . Pour tout nombre entier , on montre la convergence faible des mesures de probabilité attachées aux variables aléatoires , en calculant la distribution limite d’une manière explicite. Pour , respectivement , ce résultat mène à des formules asymptotiques pour le temps de sortie d’un billard avec des poches de rayon centrés aux coins dans un hexagone régulier, respectivement dans un carré.
Consider the region obtained by removing from the discs of radius , centered at the points of integer coordinates with . We are interested in the distribution of the free path length (exit time) of a point particle, moving from along a linear trajectory of direction , as . For every integer number , we prove the weak convergence of the probability measures associated with the random variables , explicitly computing the limiting distribution. For , respectively , this result leads to asymptotic formulas for the exit time of a billiard with pockets of radius centered at the corners and trajectory starting at the center in a regular hexagon, respectively in a square.
Classification : 11P21, 37D50, 82C40
Mots clés : Gaz de Lorentz périodique, flot linéaire, suite de Farey, réseau hexagonal
@article{AIF_2009__59_3_1043_0, author = {Boca, Florin P. and Gologan, Radu N.}, title = {On the distribution of the free path length of the linear flow in a honeycomb}, journal = {Annales de l'Institut Fourier}, pages = {1043--1075}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {59}, number = {3}, year = {2009}, doi = {10.5802/aif.2457}, mrnumber = {2543662}, zbl = {1173.37036}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2457/} }
TY - JOUR AU - Boca, Florin P. AU - Gologan, Radu N. TI - On the distribution of the free path length of the linear flow in a honeycomb JO - Annales de l'Institut Fourier PY - 2009 DA - 2009/// SP - 1043 EP - 1075 VL - 59 IS - 3 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2457/ UR - https://www.ams.org/mathscinet-getitem?mr=2543662 UR - https://zbmath.org/?q=an%3A1173.37036 UR - https://doi.org/10.5802/aif.2457 DO - 10.5802/aif.2457 LA - en ID - AIF_2009__59_3_1043_0 ER -
Boca, Florin P.; Gologan, Radu N. On the distribution of the free path length of the linear flow in a honeycomb. Annales de l'Institut Fourier, Tome 59 (2009) no. 3, pp. 1043-1075. doi : 10.5802/aif.2457. http://archive.numdam.org/articles/10.5802/aif.2457/
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