Spherical averages in the space of marked lattices

Jens Marklof*, Ilya Vinogradov

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

1 Citation (Scopus)
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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.
Original languageEnglish
Pages (from-to)75-102
Number of pages28
JournalGeometriae Dedicata
Issue number1
Early online date28 Jun 2016
Publication statusPublished - Feb 2017


  • Equidistribution
  • Homogeneous dynamics
  • Lorentz gas
  • Measure rigidity
  • Random process


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