On the reduction of a random basis
<|reference_start|>On the reduction of a random basis: For $g < n$, let $b\_1,...,b\_{n-g}$ 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 & Lov\'asz}. The proof passes by the study of the process $(r\_{g+1}^{(n)},r\_{g+2}^{(n)},...,r\_{n-1}^{(n)})$ where $r\_j^{(n)}$ is the ratio of lengths of two consecutive vectors $b^*\_{n-j+1}$ and $b^*\_{n-j}$ built from $(b\_1,...,b\_{n-g})$ 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.<|reference_end|>
@article{akhavi2006on,
    title={On the reduction of a random basis},
    author={Ali Akhavi (LIAFA), Jean-Franc{c}ois Marckert (LaBRI), Alain Rouault
  (LM-Versailles)},
    journal={arXiv preprint arXiv:math/0604331},
    year={2006},
    archivePrefix={arXiv},
    eprint={math/0604331},
    primaryClass={math.PR cs.DS}
}