Spherical averages in the space of marked lattices

Spherical averages in the space of marked lattices

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Ilya Vinogradov , Princeton University
Fine Hall 601

A marked lattice is a d-dimensional 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 faster-than-logarithmic 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 non-trivial 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 Boltzmann-Grad limit.