Abstract
A marked lattice is a d-dimensional Euclidean lattice, where each lattice point is assigned a mark via a given random field on {mathbb {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.
Highlights
Consider a Lie group G, a non-compact one-parameter subgroup R and a compact subgroup K
Let λ be a probability measure on K that is absolutely continuous with respect to Haar measure on K
Given a measure-preserving action G × X → X, (g, x) → xg on a probability space (X, A, μ), it is natural to ask under which conditions the “spherical” average Pt defined by Pt f := K f (x0k t )dλ(k) converges weakly to μ, or any other probability measure
Summary
Consider a Lie group G, a non-compact one-parameter subgroup R and a compact subgroup K. The first example of spherical equidistribution in the non-homogeneous setting for all (and not just almost all) x0 is given in [10], where the analogue of Ratner’s theorem is proved for the moduli space of branched covers of Veech surfaces, which is a fiber bundle over a homogeneous space. A major advance in this direction is the recent work by Eskin and Mirzakhani [11] and Eskin, Mirzakhani and Mohammadi [12], who prove a Ratner-like classification of measures in the moduli space of flat surfaces that are invariant under the upper triangular subgroup of SL(2, R) This is used to prove convergence of spherical averages in that moduli space, with an additional t average as above, which yields an averaged counting asymptotics for periodic trajectories in general rational billiards.
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