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

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.

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.

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)
@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 = {http://www.numdam.org/articles/10.5802/aif.2457/}
}
TY  - JOUR
AU  - Boca, Florin P.
AU  - Gologan, Radu N.
TI  - On the distribution of the free path length of the linear flow in a honeycomb
JO  - Annales de l'Institut Fourier
PY  - 2009
SP  - 1043
EP  - 1075
VL  - 59
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2457/
DO  - 10.5802/aif.2457
LA  - en
ID  - AIF_2009__59_3_1043_0
ER  - 
%0 Journal Article
%A Boca, Florin P.
%A Gologan, Radu N.
%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 http://www.numdam.org/articles/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, Tome 59 (2009) no. 3, pp. 1043-1075. doi : 10.5802/aif.2457. http://www.numdam.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 | 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.)

Cité par Sources :