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

@article{CRMATH_2017__355_6_718_0,
     author = {Cagri Sert},
     title = {Joint spectrum and large deviation principle for random matrix products},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {718--722},
     publisher = {Elsevier},
     volume = {355},
     number = {6},
     year = {2017},
     doi = {10.1016/j.crma.2017.04.015},
     language = {en},
}

TY  - JOUR
AU  - Cagri Sert
TI  - Joint spectrum and large deviation principle for random matrix products
JO  - Comptes Rendus. Mathématique
PY  - 2017
SP  - 718
EP  - 722
VL  - 355
IS  - 6
PB  - Elsevier
DO  - 10.1016/j.crma.2017.04.015
LA  - en
ID  - CRMATH_2017__355_6_718_0
ER  -

%0 Journal Article
%A Cagri Sert
%T Joint spectrum and large deviation principle for random matrix products
%J Comptes Rendus. Mathématique
%D 2017
%P 718-722
%V 355
%N 6
%I Elsevier
%R 10.1016/j.crma.2017.04.015
%G en
%F CRMATH_2017__355_6_718_0

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/

Bibliographie
Cité par

[1] H. Abels; G.A. Margulis; G.A. Soifer Semigroups containing proximal linear maps, Isr. J. Math., Volume 91 (1995) no. 1–3, pp. 1-30

[2] R.R. Bahadur Some Limit Theorems in Statistics, CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics, 1971

[3] R. Bellman Limit theorems for non-commutative operations, Duke Math. J., Volume 21 (1954) no. 3, pp. 491-500

[4] Y. Benoist Actions propres sur les espaces homogènes réductifs, Ann. Math., Volume 2 (1996), pp. 315-347

[5] Y. Benoist Propriétés asymptotiques des groupes linéaires, Geom. Funct. Anal., Volume 7 (1997) no. 1, pp. 1-47

[6] Y. Benoist; F. Labourie Sur les difféomorphismes d'Anosov affines à feuilletages stable et instable différentiables, Invent. Math., Volume 111 (1993) no. 1, pp. 285-308

[7] Y. Benoist; J.-F. Quint Random Walks on Reductive Groups, Springer International Publishing, 2016

[8] Y. Benoist; J.-F. Quint Central limit theorem for linear groups, Ann. Probab., Volume 44 (2016) no. 2, pp. 1308-1340

[9] M.A. Berger; Y. Wang Bounded semigroups of matrices, Linear Algebra Appl., Volume 166 (1992), pp. 21-27

[10] P. Bougerol; J. Lacroix Products of Random Matrices with Applications to Schrödinger Operators, Progress in Probability and Statistics, vol. 8, Birkhäuser Boston Inc., Boston, MA, USA, 1985

[11] A. Dembo; O. Zeitouni Large Deviations Techniques and Applications, vol. 38, Springer Science & Business Media, 2009

[12] H. Furstenberg Non-commuting random products, Trans. Amer. Math. Soc., Volume 108 (1963) no. 3, pp. 377-428

[13] H. Furstenberg; H. Kesten Products of random matrices, Ann. Math. Stat., Volume 31 (1960) no. 2, pp. 457-469

[14] I.Y. Goldsheid; Y. Guivarc'h Zariski closure and the dimension of the Gaussian law of the product of random matrices, I, Probab. Theory Relat. Fields, Volume 105 (1996) no. 1, pp. 109-142

[15] I.Y. Goldsheid; G. Margulis Lyapunov indices of a product of random matrices, Russ. Math. Surv., Volume 44 (1989), pp. 11-81

[16] Y. Guivarc'h On the spectrum of a large subgroup of a semisimple group, J. Mod. Dyn., Volume 2 (2008) no. 1

[17] Y. Guivarc'h; A. Raugi Frontière de Furstenberg, propriétés de contraction et théorèmes de convergence, Probab. Theory Relat. Fields, Volume 69 (1985) no. 2, pp. 187-242

[18] E. Le Page Théorèmes limites pour les produits de matrices aléatoires, Probability Measures on Groups, Springer, Berlin, Heidelberg, 1982, pp. 258-303

[19] G. Prasad $R$ -regular elements in Zariski dense subgroups, Q. J. Math., Volume 45 (1994) no. 4, pp. 542-545

[20] C. Sert Joint Spectrum and Large Deviation Principle for Random Matrix Products, Université Paris-Sud, Orsay, France, 2016 (PhD thesis)

[21] C. Sert Large deviation principle for random matrix products | arXiv

[22] C. Sert, Growth indicator on semisimple linear groups, in preparation.

[23] C. Sert, Joint spectrum in reductive groups, in preparation.

[24] C. Sert, Large deviation principle and growth indicator for Gromov hyperbolic groups, in preparation.

[25] V.N. Tutubalin A central limit theorem for products of random matrices and some of its applications, Symposia Mathematica, vol. 21, 1977, pp. 101-116

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Mesures stationnaires et fermés invariants des espaces homogènes II

Yves Benoist; Jean-François Quint

C. R. Math (2011)

Mesures stationnaires et fermés invariants des espaces homogènes

Yves Benoist; Jean-François Quint

C. R. Math (2009)

Berry–Esseen type bounds for the matrix coefficients and the spectral radius of the left random walk on $G L_{d} (ℝ)$

Christophe Cuny; Jérôme Dedecker; Florence Merlevède; ...

C. R. Math (2022)