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On the reduction of a random basis

Published online by Cambridge University Press:  22 September 2009

Ali Akhavi
Affiliation:
LIAFA, Université Denis Diderot, Case 7014, 2 place Jussieu, 75251 Paris Cedex 05, France; [email protected]
Jean-François Marckert
Affiliation:
LABRI, Université Bordeaux I, 351 cours de la Libération, 33405 Talence Cedex, France; [email protected]
Alain Rouault
Affiliation:
LMV UMR 8100, Université Versailles-Saint-Quentin, 45 avenue des États-Unis, 78035 Versailles Cedex, France; [email protected]
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Abstract

For p ≤ n, let b1(n),...,bp(n) be independent random vectors in $\mathbb{R}^n$ 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 $\widehat b_{1}^{(n)},\ldots, \widehat b_p^{(n)}$ is the basis obtained from b1(n),...,bp(n) by Gram-Schmidt orthogonalization, the quality of the reduction depends upon the sequence of ratios of squared lengths of consecutive vectors $r_j^{(n)} = \Vert \widehat b^{(n)}_{n-j+1}\Vert^2 / \Vert \widehat b^{(n)}_{n-j} \Vert^2$ , j = 1,...,p - 1. We show that as n → +∡ the process $(r_j^{(n)}-1,j\geq 1)$ tends in distribution in some sense to an explicit process $({\mathcal R}_j -1,j\geq 1)$ ; some properties of the latter are provided. The probability that a random random basis is s-LLL-reduced is then showed to converge for p = n - g, and g fixed, or g = g(n) → +∞.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2009

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