On the reduction of a random basis
Résumé
For $g < n$, let $b_1,\dots,b_{n-g}$ be $n - g$ independent vectors in $\mathbb{R}^n$ 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 $n\to +\infty$ concerning the property that such a random basis is reduced in the sense of {\sc Lenstra, Lenstra \& Lovász}. \\ The proof passes by the study of the process $(r_{g+1}^{(n)},r_{g+2}^{(n)},\dots,r_{n-1}^{(n)})$ where $r_j^{(n)}$ is the ratio of lengths of two consecutive vectors $b^*_{n-j+1}$ and $b^*_{n-j}$ built from $(b_1,\dots,b_{n-g})$ by the Gram--Schmidt orthogonalization procedure, which we believe to be interesting in its own. We show that, as $n\to+\infty$, the process $(r_j^{(n)}-1)_j$ tends in distribution in some sense to an explicit process $({\mathcal R}_j -1)_j$; some properties of this latter are provided.
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