Spherical averages in the space of marked lattices

TL;DR

This paper proves that large spheres in marked lattices with strongly mixing random marks become equidistributed, and applies this to analyze the distribution of free path lengths in crystals with defects.

Contribution

It establishes equidistribution results for marked lattices with strongly mixing random fields and explores their implications for crystal defect models.

Findings

Large spheres become equidistributed in the space of marked lattices.

The free path length in crystals with random defects has a limiting distribution.

The space of marked lattices forms a non-trivial fiber bundle, not a homogeneous space.

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.