On the distribution of the free path length of the linear flow in a honeycomb

Nous considérons la région obtenue en enlevant de ${ℝ}^2$ les disques de rayon $\epsilon$, centrés aux points de coordonnées entières $(a,b)$ avec $b\not\equiv a(mod\ell )$. 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 $(0,0)$ 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.

Consider the region obtained by removing from ${ℝ}^2$ the discs of radius $\epsilon$, centered at the points of integer coordinates $(a,b)$ with $b\not\equiv a(mod\ell )$. 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 $(0,0)$ 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.

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