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On the probability of a random lattice avoiding a large convex set
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
2011 (English)In: Proceedings of the London Mathematical Society, ISSN 0024-6115, E-ISSN 1460-244X, Vol. 103, no 6, 950-1006 p.Article in journal (Refereed) Published
Abstract [en]

Given a set ℭ ⊂ ℝd, 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) that 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 p0. 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.

Place, publisher, year, edition, pages
2011. Vol. 103, no 6, 950-1006 p.
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URN: urn:nbn:se:uu:diva-140261DOI: 10.1112/plms/pdr021ISI: 000297910200002OAI: oai:DiVA.org:uu-140261DiVA: diva2:383324
Available from: 2011-01-04 Created: 2011-01-04 Last updated: 2015-07-23Bibliographically approved

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Strömbergsson, Andreas
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Analysis and Applied Mathematics
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