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

Let $\mathrm{\ell }\ge 2$ be an integer. For each $\text{eps}>0$ remove from ${\mathbb{R}}^2$ the union of discs of radius $\text{eps}$ centered at the integer lattice points $(m,n$, with $m\text{nequiv}n\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{\ell }$. 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 ${\tau }_{\mathrm{\ell },\text{eps}}(\omega )\in [0,\mathrm{\infty }]$. We prove the weak convergence of the probability measures associated with the random variables $\text{eps}{\tau }_{\mathrm{\ell },\text{eps}}$ as $\text{eps}\to 0^{+}$ and explicitly compute the limiting distribution. For $\mathrm{\ell }=3$ 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 $\text{eps}\to 0^{+}$ removed from the corners. For $\mathrm{\ell }=2$ 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 $[0,\mathrm{\infty })$ 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.