Spherical averages in the space of marked lattices

Jens Marklof; Ilya Vinogradov

doi:10.1007/s10711-016-0180-2

Spherical averages in the space of marked lattices

Jens Marklof^*
, Ilya Vinogradov

^*Corresponding author for this work

Research output: Contribution to journal › Article (Academic Journal) › peer-review

4 Citations (Scopus)

407 Downloads (Pure)

Abstract

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.

Original language	English
Pages (from-to)	75-102
Number of pages	28
Journal	Geometriae Dedicata
Volume	186
Issue number	1
Early online date	28 Jun 2016
DOIs	https://doi.org/10.1007/s10711-016-0180-2
Publication status	Published - Feb 2017

Keywords

Equidistribution
Homogeneous dynamics
Lorentz gas
Measure rigidity
Random process

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AB - A marked lattice is a d-dimensional Euclidean lattice, where each lattice point is assigned a mark via a given random field on Zd. 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.

KW - Equidistribution

KW - Homogeneous dynamics

KW - Lorentz gas

KW - Measure rigidity

KW - Random process

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SP - 75

EP - 102

JO - Geometriae Dedicata

JF - Geometriae Dedicata

IS - 1

ER -

Spherical averages in the space of marked lattices

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Spherical averages in the space of marked lattices

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