On the distribution of the free path length of the linear flow in a honeycomb

Florin P. Boca, Radu N. Gologan

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Pages (from-to)1043-1075
Number of pages33
JournalAnnales de l'Institut Fourier
Issue number3
StatePublished - 2009


  • Farey fractions
  • Honeycomb lattice
  • Linear flow
  • Periodic lorentz gas

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Geometry and Topology


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