On the Reduction of a Random Basis

For g < n, let b1, . . . , bn−g be n− g independent vectors in R with a common distribution invariant by rotation. Considering these vectors as a basis for the Euclidean lattice they generate, the aim of this paper is to provide asymptotic results when n → +∞ concerning the property that such a random basis is reduced in the sense of Lenstra, Lenstra & Lovász. The proof passes by the study of the process (r (n) g+1, r (n) g+2, . . . , r (n) n−1) where r (n) j is the ratio of lengths of two consecutive vectors b∗n−j+1 and b ∗ n−j built from (b1, . . . , bn−g) by the Gram– Schmidt orthogonalization procedure, which we believe to be interesting in its own. We show that, as n → +∞, the process (r j − 1)j tends in distribution in some sense to an explicit process (Rj − 1)j; some properties of this latter are provided.