On the reduction of a random basis
ESAIM: Probability and Statistics, Tome 13 (2009), pp. 437-458.

For pn, let b 1 (n) ,...,b p (n) be independent random vectors in 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 b ^ 1 (n) ,...,b ^ p (n) is the basis obtained from b 1 (n) ,...,b p (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) =b ^ n-j+1 (n) 2 /b ^ n-j (n) 2 , j=1,...,p-1. We show that as n the process (r j (n) -1,j1) tends in distribution in some sense to an explicit process ( j -1,j1); 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)+.

DOI : 10.1051/ps:2008012
Classification : 15A52, 15A03, 60B12, 60F99, 06B99, 68W40
Mots clés : random matrices, random basis, orthogonality index, determinant, lattice reduction
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Akhavi, Ali; Marckert, Jean-François; Rouault, Alain. On the reduction of a random basis. ESAIM: Probability and Statistics, Tome 13 (2009), pp. 437-458. doi : 10.1051/ps:2008012. http://www.numdam.org/articles/10.1051/ps:2008012/

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