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

Aoun, Richard; Mathieu, Pierre; Sert, Cagri

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

Let $(X,d)$ be a separable geodesic Gromov-hyperbolic space, $o\in X$ a basepoint and $\mu$ a countably supported non-elementary probability measure on $\mathrm{Isom}\left(X\right)$. Denote by $z_n$ the random walk on $X$ driven by the probability measure $\mu$. Supposing that $\mu$ 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.

Soit $(X,d)$ un espace Gromov-hyperbolique, géodésique et séparable, $o\in X$ un point base et $\mu$ une mesure de probabilité non élémentaire et à support dénombrable sur le groupe $\mathrm{Isom}\left(X\right)$ des isométries de $X$. Notons par $z_n$ la marche aléatoire sur $X$ induite par $\mu$. Sous l’hypothèse de moment exponentiel fini de $\mu$, 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.

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

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