Random walks on hyperbolic spaces: second order expansion of the rate function at the drift

Richard Aoun; Pierre Mathieu; Cagri Sert

doi:10.5802/jep.225

Random walks on hyperbolic spaces: second order expansion of the rate function at the drift
[Marches aléatoires sur les espaces hyperboliques : dérivée seconde en la vitesse de fuite de la fonction de taux des grandes déviations]

Richard Aoun ¹ ; Pierre Mathieu ² ; Cagri Sert ³

¹ University Gustave Eiffel, Champs-sur-Marne 5 boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée Cedex 2, France
² Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373 13453 Marseille, France
³ Institut für Mathematik, Universität Zürich 190, Winterthurerstrasse, 8057 Zürich, Switzerland

Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 549-573.

Résumé
Abstract

Soit $(X, d)$ un espace Gromov-hyperbolique, géodésique et séparable, $o \in X$ un point base et $μ$ une mesure de probabilité non élémentaire et à support dénombrable sur le groupe $Isom (X)$ des isométries de $X$ . Notons par $z_{n}$ la marche aléatoire sur $X$ induite par $μ$ . Sous l’hypothèse de moment exponentiel fini de $μ$ , nous donnons un développement de Taylor d’ordre $2$ de la fonction de taux des grandes déviations de la suite de variables aléatoires $\frac{1}{n} d (z_{n}, o)$ et exprimons la dérivée seconde en la vitesse de fuite en fonction de la variance dans le théorème central limite que vérifie la suite $d (z_{n}, o)$ . Cela répond par l’affirmative à une question posée dans [6]. La preuve s’appuie sur l’étude de la transformée de Laplace de $d (z_{n}, o)$ en zéro en utilisant une approximation par une martingale introduite pour la première fois par Benoist-Quint, combinée avec une transformée exponentielle de martingales et des estimées de grandes déviations pour le crochet de certaines martingales.

Let $(X, d)$ be a separable geodesic Gromov-hyperbolic space, $o \in X$ a basepoint and $μ$ a countably supported non-elementary probability measure on $Isom (X)$ . Denote by $z_{n}$ the random walk on $X$ driven by the probability measure $μ$ . Supposing that $μ$ has a finite exponential moment, we give a second-order Taylor expansion of the large deviation rate function of the sequence $\frac{1}{n} d (z_{n}, o)$ and show that the corresponding coefficient is expressed by the variance in the central limit theorem satisfied by the sequence $d (z_{n}, o)$ . This provides a positive answer to a question raised in [6]. The proof relies on the study of the Laplace transform of $d (z_{n}, o)$ at the origin using a martingale decomposition first introduced by Benoist–Quint together with an exponential submartingale transform and large deviation estimates for the quadratic variation process of certain martingales.

Reçu le : 2022-04-20
Accepté le : 2022-12-20
Publié le : 2023-03-20

DOI : 10.5802/jep.225

Classification : 20F67, 60G50, 60G42, 60F10, 60F05
Keywords: Random walks, hyperbolic spaces, martingales, large deviations, central limit theorem
Mot clés : Marches aléatoires, espaces hyperboliques, martingales, grandes déviations, théorème central limite

Affiliations des auteurs :

Richard Aoun ¹ ; Pierre Mathieu ² ; Cagri Sert ³

¹ University Gustave Eiffel, Champs-sur-Marne 5 boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée Cedex 2, France
² Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373 13453 Marseille, France
³ Institut für Mathematik, Universität Zürich 190, Winterthurerstrasse, 8057 Zürich, Switzerland

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Richard Aoun and Pierre Mathieu and Cagri Sert},
     title = {Random walks on hyperbolic spaces: second order expansion of the rate function at the drift},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {549--573},
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Richard Aoun; Pierre Mathieu; Cagri Sert. Random walks on hyperbolic spaces: second order expansion of the rate function at the drift. Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 549-573. doi : 10.5802/jep.225. https://jep.centre-mersenne.org/articles/10.5802/jep.225/

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