no. 3
On the distribution of the free path length of the linear flow in a honeycomb
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, Volume 59 (2009) no. 3, pp. 1043-1075.

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.

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.

Received:
Revised:
Accepted:
DOI: 10.5802/aif.2457
Classification: 11P21,  37D50,  82C40
Keywords: Periodic Lorentz gas, linear flow, Farey fractions, honeycomb lattice
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) 
@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},
     pages = {1043--1075},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {59},
     number = {3},
     year = {2009},
     doi = {10.5802/aif.2457},
     zbl = {1173.37036},
     mrnumber = {2543662},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2457/}
}
TY  - JOUR
TI  - On the distribution of the free path length of the linear flow in a honeycomb
JO  - Annales de l'Institut Fourier
PY  - 2009
DA  - 2009///
SP  - 1043
EP  - 1075
VL  - 59
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2457/
UR  - https://zbmath.org/?q=an%3A1173.37036
UR  - https://www.ams.org/mathscinet-getitem?mr=2543662
UR  - https://doi.org/10.5802/aif.2457
DO  - 10.5802/aif.2457
LA  - en
ID  - AIF_2009__59_3_1043_0
ER  - 
%0 Journal Article
%T On the distribution of the free path length of the linear flow in a honeycomb
%J Annales de l'Institut Fourier
%D 2009
%P 1043-1075
%V 59
%N 3
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2457
%R 10.5802/aif.2457
%G en
%F 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/articles/10.5802/aif.2457/

[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 | Article | MR: 1785463 | Zbl: 0989.11049

[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: 2028172 | Zbl: 1066.37021

[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 | Article | MR: 2004979 | Zbl: 1078.37006

[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 | Article | MR: 2186997 | Zbl: 1154.11022

[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 | Article | MR: 2274553 | Zbl: 1143.37002

[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 | Article | MR: 1981109 | Zbl: 1041.82016

[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: 2417573 | Zbl: 1145.82019

[8] Dahlqvist, P. The Lyapunov exponent in the Sinai billiard in the small scatterer limit, Nonlinearity, Volume 10 (1997) no. 1, pp. 159-173 | Article | MR: 1430746 | Zbl: 0907.58038

[9] Golse, F. The periodic Lorentz gas in the Boltzman-Grad limit, International Congress of Mathematicians, Volume III (2006), pp. 183-201 | MR: 2275676 | Zbl: 1109.82020

[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.)

Cited by Sources: