The statistics of particle trajectories in the inhomogeneous Sinai problem for a two-dimensional lattice

In connection with the two-dimensional model known as the ‘periodic Lorentz gas’, we study the asymptotic behaviour of statistical characteristics of a free path interval of a point particle before its first occurrence in an $h$-neighbourhood (a circle of radius $h$) of a non-zero integer point as $h\to 0$ given that the particle starts from the $h$-neighbourhood of the origin. We evaluate the limit distribution function of the free path length and of the input aimed parameter (the distance from the trajectory to the integer point we are interested in) for a given value of the output aimed parameter. This problem was studied earlier for a particle starting from the origin (the homogeneous case).