On the reduction of a random basis

International audience 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