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

Boca, Florin P.; Gologan, Radu N.

Annales de l'Institut Fourier, Volume 59 (2009) no. 3, pp. 1043-1075.

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.

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.

[1] Boca, F. P.; Cobeli, C.; Zaharescu, A. Distribution of lattice points visible from the origin, Comm. Math. Phys., Volume 213 (2000) no. 2, pp. 433-470 | DOI | MR | Zbl

[2] Boca, F. P.; Gologan, R. N.; Zaharescu, A. The average length of a trajectory in a certain billiard in a flat two-torus, New York J. Math. (electronic), Volume 9 (2003), pp. 303-330 | MR | Zbl

[3] Boca, F. P.; Gologan, R. N.; Zaharescu, A. The statistics of the trajectory of a certain billiard in a flat two-torus, Comm. Math. Phys., Volume 240 (2003) no. 1-2, pp. 53-73 | DOI | MR | Zbl

[4] Boca, F. P.; Zaharescu, A. On the correlations of directions in the Euclidean plane, Trans. Amer. Math. Soc., Volume 358 (2006) no. 4, pp. 1797-1825 | DOI | MR | Zbl

[5] Boca, F. P.; Zaharescu, A. The distribution of the free path lengths in the periodic two-dimensional Lorentz gas in the small-scatterer limit, Comm. Math. Phys., Volume 269 (2007) no. 2, pp. 425-471 | DOI | MR | Zbl

[6] Caglioti, E.; Golse, F. On the distribution of free path lengths for the periodic Lorentz gas. III., Comm. Math. Phys., Volume 236 (2003) no. 2, pp. 199-221 | DOI | MR | Zbl

[7] Caglioti, E.; Golse, F. The Boltzman-Grad limit of the periodic Lorentz gas in two space dimensions, C. R. Math. Acad. Sci. Paris, Volume 346 (2008) no. 7-8, pp. 477-482 | MR | Zbl

[8] Dahlqvist, P. The Lyapunov exponent in the Sinai billiard in the small scatterer limit, Nonlinearity, Volume 10 (1997) no. 1, pp. 159-173 | DOI | MR | Zbl

[9] Golse, F. The periodic Lorentz gas in the Boltzman-Grad limit, International Congress of Mathematicians, Volume III, Eur. Math. Soc., Zürich (2006), pp. 183-201 | MR | Zbl

[10] Marklof, J.; Strömbergsson, A. The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems (to appear in Ann. of Math.)