Joint spectrum and large deviation principle for random matrix products

Sert, Cagri

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

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 Sn={g1.….gn|gi∈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 Sn={g1.….gn|gi∈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.

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