The Distribution of the Linear Flow Length in a Honeycomb in the Small-scatterer Limit

From the regular hexagon of unit size remove circular holes of small radius ε > 0 centered at the vertices, obtaining a region Hε of area 3 √ 3 2 − πε2. For each pair (x, ω) ∈ Hε × [0, 2π] consider a point particle moving at unit speed on a linear trajectory with specular reflections when meeting the boundary. Denote by τhex ε (x, ω) the time it takes the particle to reach one of the holes. This quantity is called the free path length (or first exit time). Equivalently, one can think about “fat points” (obstacles or scatterers) of radius ε in the unit honeycomb lattice in R2, and of a particle moving at unit speed and velocity ω on a linear trajectory until it hits one of the obstacles. If the initial position x is always chosen in a fundamental domain, the first hitting time coincides with τhex ε (x, ω). In this paper we are interested in estimating the probability