On the reduction of a random basis
| dc.creator | Akhavi, Ali | |
| dc.creator | Marckert, Jean-François | |
| dc.creator | Rouault, Alain | |
| dc.date | 2006-04-14 | |
| dc.date.accessioned | 2026-07-07T07:10:56Z | |
| dc.date.available | 2026-07-07T07:10:56Z | |
| dc.description | 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 & Lovász}. 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. | |
| dc.identifier | https://arxiv.org/abs/math/0604331 | |
| dc.identifier | http://arxiv.org/abs/math/0604331 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/111575 | |
| dc.subject | Probability | |
| dc.subject | Data Structures and Algorithms | |
| dc.title | On the reduction of a random basis | |
| dc.type | text |