Spherical averages in the space of marked lattices
Spherical averages in the space of marked lattices

Ilya Vinogradov , Princeton University
Fine Hall 601
A marked lattice is a ddimensional Euclidean lattice, where each lattice point is assigned a mark via a given random field on Z^d. We prove that, if the field is strongly mixing with a fasterthanlogarithmic rate, then for every given lattice and almost every marking, large spheres become equidistributed in the space of marked lattices. A key aspect of our study is that the space of marked lattices is not a homogeneous space, but rather a nontrivial fiber bundle over such a space. As an application, we prove that the free path length in a crystal with random defects has a limiting distribution in the BoltzmannGrad limit.