no. 3
On the distribution of the free path length of the linear flow in a honeycomb
[Sur la distribution du temps de sortie pour le flot linéaire dans un réseau hexagonal]
Boca, Florin P.1 ; Gologan, Radu N.2
1 University of Illinois at Urbana-Champaign Department of Mathematics 1409 W. Green St. Urbana, IL 61801 (USA)
2 Institute of Mathematics of the Romanian Academy P.O.Box 1-764 Bucharest 014700 (Romania)
Annales de l'Institut Fourier, Tome 59 (2009) no. 3, pp. 1043-1075.

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

DOI : 10.5802/aif.2457
Classification : 11P21, 37D50, 82C40
Keywords: Periodic Lorentz gas, linear flow, Farey fractions, honeycomb lattice
Mot clés : Gaz de Lorentz périodique, flot linéaire, suite de Farey, réseau hexagonal

Boca, Florin P. 1 ; Gologan, Radu N. 2

1 University of Illinois at Urbana-Champaign Department of Mathematics 1409 W. Green St. Urbana, IL 61801 (USA)
2 Institute of Mathematics of the Romanian Academy P.O.Box 1-764 Bucharest 014700 (Romania) 
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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, Tome 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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