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- W1862627583 abstract "Let $ell geq 2$ be an integer. For each $eps >0$ remove from $R^2$ the union of discs of radius $eps$ centered at the integer lattice points $(m,n$, with $mnequiv nmod{ell}$. Consider a point-like particle moving linearly at unit speed, with velocity $omega$, along a trajectory starting at the origin, and its free path length $tau_{ell,eps} (omega)in [0,infty]$. We prove the weak convergence of the probability measures associated with the random variables $eps tau_{ell,eps}$ as $eps to 0^+$ and explicitly compute the limiting distribution. For $ell=3$ this leads to an asymptotic formula for the length of the trajectory of a billiard in a regular hexagon, starting at the center, with circular pockets of radius $epsto 0^+$ removed from the corners. For $ell=2$ this corresponds to the trajectory of a billiard in a unit square with circular pockets removed from the corners and trajectory starting at the center of the square. The limiting probability measures on $[0,infty)$ have a tail at infinity, which contrasts with the case of a square with pockets and trajectory starting from one of the corners, where the limiting probability measure has compact support." @default.
- W1862627583 created "2016-06-24" @default.
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- W1862627583 date "2008-02-07" @default.
- W1862627583 modified "2023-09-27" @default.
- W1862627583 title "On the distribution of the free path length of the linear flow in a honeycomb" @default.
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