Abstract
Consider the region obtained by removing from ℝ2 the discs of radius ε, centered at the points of integer coordinates (a, b) with b ≢ a (mod ℓ). We are interested in the distribution of the free path length (exit time) τℓε(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 ετℓ,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.
Original language | English (US) |
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Pages (from-to) | 1043-1075 |
Number of pages | 33 |
Journal | Annales de l'Institut Fourier |
Volume | 59 |
Issue number | 3 |
DOIs | |
State | Published - 2009 |
Keywords
- Farey fractions
- Honeycomb lattice
- Linear flow
- Periodic lorentz gas
ASJC Scopus subject areas
- Algebra and Number Theory
- Geometry and Topology