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.
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.
Revised:
Accepted:
DOI: 10.5802/aif.2457
Classification: 11P21, 37D50, 82C40
Keywords: Periodic Lorentz gas, linear flow, Farey fractions, honeycomb lattice
Author's affiliations:
@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}, zbl = {1173.37036}, mrnumber = {2543662}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2457/} }
TY - JOUR 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 - https://aif.centre-mersenne.org/articles/10.5802/aif.2457/ UR - https://zbmath.org/?q=an%3A1173.37036 UR - https://www.ams.org/mathscinet-getitem?mr=2543662 UR - https://doi.org/10.5802/aif.2457 DO - 10.5802/aif.2457 LA - en ID - AIF_2009__59_3_1043_0 ER -
%0 Journal Article %T On the distribution of the free path length of the linear flow in a honeycomb %J Annales de l'Institut Fourier %D 2009 %P 1043-1075 %V 59 %N 3 %I Association des Annales de l’institut Fourier %U https://doi.org/10.5802/aif.2457 %R 10.5802/aif.2457 %G en %F AIF_2009__59_3_1043_0
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, Volume 59 (2009) no. 3, pp. 1043-1075. doi : 10.5802/aif.2457. https://aif.centre-mersenne.org/articles/10.5802/aif.2457/
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