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 2 the discs of radius ε, centered at the points of integer coordinates (a,b) with ba(mod). We are interested in the distribution of the free path length (exit time) τ ,ε (ω) of a point particle, moving from (0,0) along a linear trajectory of direction ω, as ε0 + . For every integer number 2, we prove the weak convergence of the probability measures associated with the random variables ετ ,ε , explicitly computing the limiting distribution. For =3, respectively =2, this result leads to asymptotic formulas for the exit time of a billiard with pockets of radius ε0 + 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 2 les disques de rayon ε, centrés aux points de coordonnées entières (a,b) avec ba(mod). Nous étudions la répartition de la longueur du libre parcours (temps de sortie) τ ,ε (ω) d’une particule ponctuelle, partant de (0,0) sur une trajectoire rectiligne de direction ω quand ε0 + . Pour tout nombre entier 2, on montre la convergence faible des mesures de probabilité attachées aux variables aléatoires ετ ,ε , en calculant la distribution limite d’une manière explicite. Pour =3, respectivement =2, ce résultat mène à des formules asymptotiques pour le temps de sortie d’un billard avec des poches de rayon ε0 + 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}
}
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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