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

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 $m\nequiv n\mod{\ell}$. 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 $\eps\to 0^+$ 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.