Joint spectrum and large deviation principle for random matrix products

Cagri Sert

doi:10.1016/j.crma.2017.04.015

Probability theory/Geometry

Joint spectrum and large deviation principle for random matrix products
[Spectre joint et principe de grandes déviations pour les produits de matrices aléatoires]

Présenté par : Jean-François Le Gall

Cagri Sert ¹

¹ Departement Mathematik, ETHZ, Rämistrasse 101, CH-8092 Zürich, Switzerland

Comptes Rendus. Mathématique, Volume 355 (2017) no. 6, pp. 718-722.

Résumé
Abstract

Le but de cette note est d'énoncer certains résultats sur les propriétés asymptotiques probabilistes et déterministes des groupes linéaires. Le premier est l'homologue, pour les normes des produits de matrices aléatoires, du théorème classique de Cramér sur le principe de grandes déviations des sommes des variables aléatoires iid. Dans le deuxième résultat, nous introduisons un ensemble limite décrivant la forme asymptotique des puissances $S^{n} = {g_{1} . \dots . g_{n} | g_{i} \in S}$ d'une partie S d'un groupe de Lie linéaire semisimple (e.g., $SL (d, R)$ ). Cet ensemble limite trouve, parmi d'autres, une application dans l'étude des grandes déviations.

The aim of this note is to announce some results about the probabilistic and deterministic asymptotic properties of linear groups. The first one is the analogue, for norms of random matrix products, of the classical theorem of Cramér on large deviation principles (LDP) for sums of iid real random variables. In the second result, we introduce a limit set describing the asymptotic shape of the powers $S^{n} = {g_{1} . \dots . g_{n} | g_{i} \in S}$ of a subset S of a semisimple linear Lie group G (e.g., $SL (d, R)$ ). This limit set has applications, among others, in the study of large deviations.

Reçu le : 2017-02-10
Accepté le : 2017-04-28
Publié le : 2017-05-09

DOI : 10.1016/j.crma.2017.04.015

Affiliations des auteurs :

Cagri Sert ¹

¹ Departement Mathematik, ETHZ, Rämistrasse 101, CH-8092 Zürich, Switzerland

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Cagri Sert. Joint spectrum and large deviation principle for random matrix products. Comptes Rendus. Mathématique, Volume 355 (2017) no. 6, pp. 718-722. doi : 10.1016/j.crma.2017.04.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.04.015/

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