# ANNALES DE L'INSTITUT FOURIER

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, p. 1043-1075
Consider the region obtained by removing from ${ℝ}^{2}$ the discs of radius $\epsilon$, centered at the points of integer coordinates $\left(a,b\right)$ with $b\not\equiv a\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}\ell \right)$. We are interested in the distribution of the free path length (exit time) ${\tau }_{\ell ,\epsilon }\left(\omega \right)$ of a point particle, moving from $\left(0,0\right)$ along a linear trajectory of direction $\omega$, as $\epsilon \to {0}^{+}$. For every integer number $\ell \ge 2$, we prove the weak convergence of the probability measures associated with the random variables $\epsilon {\tau }_{\ell ,\epsilon }$, explicitly computing the limiting distribution. For $\ell =3$, respectively $\ell =2$, this result leads to asymptotic formulas for the exit time of a billiard with pockets of radius $\epsilon \to {0}^{+}$ 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 ${ℝ}^{2}$ les disques de rayon $\epsilon$, centrés aux points de coordonnées entières $\left(a,b\right)$ avec $b\not\equiv a\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}\ell \right)$. Nous étudions la répartition de la longueur du libre parcours (temps de sortie) ${\tau }_{\ell ,\epsilon }\left(\omega \right)$ d’une particule ponctuelle, partant de $\left(0,0\right)$ sur une trajectoire rectiligne de direction $\omega$ quand $\epsilon \to {0}^{+}$. Pour tout nombre entier $\ell \ge 2$, on montre la convergence faible des mesures de probabilité attachées aux variables aléatoires $\epsilon {\tau }_{\ell ,\epsilon }$, en calculant la distribution limite d’une manière explicite. Pour $\ell =3$, respectivement $\ell =2$, ce résultat mène à des formules asymptotiques pour le temps de sortie d’un billard avec des poches de rayon $\epsilon \to {0}^{+}$ centrés aux coins dans un hexagone régulier, respectivement dans un carré.
DOI : https://doi.org/10.5802/aif.2457
Classification:  11P21,  37D50,  82C40
Keywords: Periodic Lorentz gas, linear flow, Farey fractions, honeycomb lattice
@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},
publisher = {Association des Annales de l'institut Fourier},
volume = {59},
number = {3},
year = {2009},
pages = {1043-1075},
doi = {10.5802/aif.2457},
zbl = {1173.37036},
mrnumber = {2543662},
language = {en},
url = {https://aif.centre-mersenne.org/item/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/item/AIF_2009__59_3_1043_0/

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