On the probability of a random lattice avoiding a large convex set
Abstract
Given a set C in R^d, let p(C) be the probability that a random d-dimensional unimodular lattice, chosen according to Haar measure on SL(d,Z)\SL(d,R), is disjoint from C\{0}. For special convex sets C we prove bounds on p(C) which are sharp up to a scaling of C by a constant. We also prove bounds on a variant of p(C) where the probability is conditioned on the random lattice containing a fixed given point p. Our bounds have applications, among other things, to the asymptotic properties of the collision kernel of the periodic Lorentz gas in the Boltzmann-Grad limit, in arbitrary dimension d.
Keywords
Cite
@article{arxiv.1008.3805,
title = {On the probability of a random lattice avoiding a large convex set},
author = {Andreas Strömbergsson},
journal= {arXiv preprint arXiv:1008.3805},
year = {2014}
}
Comments
53 pages, 6 Figures. The introduction has been extended and all the figures added, prompted by suggestions of the referee. The paper has now been accepted and will appear in Proceedings of the London Mathematical Society. (However the published version will differ slightly from the present version.)