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Article Dans Une Revue ESAIM: Probability and Statistics Année : 2009

On the reduction of a random basis

Résumé


For , let be independent random vectors in with the same distribution invariant by rotation and without mass at the origin. Almost surely these vectors form a basis for the Euclidean lattice they generate. The topic of this paper is the property of reduction of this random basis in the sense of Lenstra-Lenstra-Lovász (LLL). If is the basis obtained from by Gram-Schmidt orthogonalization, the quality of the reduction depends upon the sequence of ratios of squared lengths of consecutive vectors , . We show that as the process tends in distribution in some sense to an explicit process ; some properties of the latter are provided. The probability that a random random basis is -LLL-reduced is then showed to converge for , and fixed, or .

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Dates et versions

hal-00520007 , version 1 (22-09-2010)

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Ali Akhavi, Jean-François Marckert, Alain Rouault. On the reduction of a random basis. ESAIM: Probability and Statistics, 2009, 13, pp.437-458. ⟨10.1051/ps:2008012⟩. ⟨hal-00520007⟩
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