3 research outputs found
On the reduction of a random basis
For , let be independent vectors in
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 concerning the
property that such a random basis is reduced in the sense of {\sc Lenstra,
Lenstra & Lov\'asz}. The proof passes by the study of the process
where is the
ratio of lengths of two consecutive vectors and
built from by the Gram--Schmidt orthogonalization
procedure, which we believe to be interesting in its own. We show that, as
, the process tends in distribution in some
sense to an explicit process ; some properties of this
latter are provided
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