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A marked lattice is a $d$-dimensional Euclidean lattice, where each latticepoint is assigned a mark via a given random field on ${\mathbb Z}^d$. We provethat, if the field is strongly mixing with a faster-than-logarithmic rate, thenfor every given lattice and almost every marking, large spheres becomeequidistributed in the space of marked lattices. A key aspect of our study isthat the space of marked lattices is not a homogeneous space, but rather anon-trivial fiber bundle over such a space. As an application, we prove thatthe free path length in a crystal with random defects has a limitingdistribution in the Boltzmann-Grad limit.

Spherical averages in the space of marked lattices