## Abstract

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 (mod ℓ). We are interested in the distribution of the free path length (exit time) τ_{ℓε}(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.

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