Central limit theorem on $\mathrm{CAT}(0)$ spaces with contracting isometries
Le Bars, Corentin1
1Weizmann Institute of Science, Rehovot (Israel)
Annales de l'Institut Fourier, Online first, 42 p.

Let $G$ be a group acting on a $\mathrm{CAT}(0)$ space with contracting isometries. We study the random walk generated by an admissible measure on $G$. We prove that if the action is non-elementary and under optimal moment assumptions on the measure, the random walk satisfies a central limit theorem. The general approach is inspired from the cocycle argument of Y. Benoist and J-F. Quint, and our strategy relies on the use of hyperbolic models introduced by H. Petyt, A. Zalloum and D. Spriano, which are analogues of the contact graph for the class of $\mathrm{CAT}(0)$ spaces. As a side result, we prove that the probability that the $n$th-step the random walk acts as a contracting isometry goes to $1$ as $n$ goes to infinity.

Soit $G$ un groupe agissant sur un espace $\mathrm{CAT}(0)$ avec des isométries contractantes. On étudie une marche aléatoire engendrée par une mesure admissible sur $G$ et on prouve, sous des hypothèse optimales de moment, que la marche aléatoire satisfait un théorème de la limite centrale. L’approche générale est inspirée d’un argument sur les cocycles dû à Y. Benoist et J-F. Quint, et notre stratégie repose sur l’utilisation de modèles hyperboliques pour les espaces $\mathrm{CAT}(0)$ introduits par H. Petyt, A. Zalloum et D. Spriano, une construction analogue au graphe de contact pour les complexes cubiques $\mathrm{CAT}(0)$. Nous prouvons également que la probabilité que le $n$-ième pas de la marche aléatoire soit une isométrie contractante tend vers $1$ lorsque $n$ tend vers $+\infty $.

Received:
Revised:
Accepted:
Online First:
DOI: 10.5802/aif.3717
Classification: 20F65, 37A15, 60B15, 22D40
Keywords: Geometric group theory, random walks, central limit theorem, non-positive curvature
Mots-clés : Théorie géométrique des groupes, Marches aléatoires, Théorème central limite, courbure non-positive

Le Bars, Corentin  1

1 Weizmann Institute of Science, Rehovot (Israel) 
@unpublished{AIF_0__0_0_A5_0,
     author = {Le Bars, Corentin},
     title = {Central limit theorem on $\mathrm{CAT}(0)$ spaces with contracting isometries},
     journal = {Annales de l'Institut Fourier},
     year = {2025},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     doi = {10.5802/aif.3717},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - Le Bars, Corentin
TI  - Central limit theorem on $\mathrm{CAT}(0)$ spaces with contracting isometries
JO  - Annales de l'Institut Fourier
PY  - 2025
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3717
LA  - en
ID  - AIF_0__0_0_A5_0
ER  - 
%0 Unpublished Work
%A Le Bars, Corentin
%T Central limit theorem on $\mathrm{CAT}(0)$ spaces with contracting isometries
%J Annales de l'Institut Fourier
%D 2025
%V 0
%N 0
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3717
%G en
%F AIF_0__0_0_A5_0
Le Bars, Corentin. Central limit theorem on $\mathrm{CAT}(0)$ spaces with contracting isometries. Annales de l'Institut Fourier, Online first, 42 p.

[1] Ballmann, Werner Lectures on spaces of nonpositive curvature, DMV Seminar, 25, Birkhäuser, 1995 (with an appendix by Misha Brin) | Zbl | DOI | MR

[2] Benoist, Yves; Quint, Jean-François Central limit theorem for linear groups, Ann. Probab., Volume 44 (2016) no. 2, pp. 1308-1340 | Zbl | MR | DOI

[3] Benoist, Yves; Quint, Jean-François Central limit theorem on hyperbolic groups, Izv. Ross. Akad. Nauk, Ser. Mat., Volume 80 (2016) no. 1, pp. 3-23 | Zbl | DOI | MR

[4] Bestvina, Mladen; Fujiwara, Koji A characterization of higher rank symmetric spaces via bounded cohomology, Geom. Funct. Anal., Volume 19 (2009) no. 1, pp. 11-40 | Zbl | DOI | MR

[5] Björklund, Michael Central limit theorems for Gromov hyperbolic groups, J. Theor. Probab., Volume 23 (2010) no. 3, pp. 871-887 | Zbl | DOI | MR

[6] Bonk, Mario; Schramm, Oded Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal., Volume 10 (2000) no. 2, pp. 266-306 | Zbl | DOI | MR

[7] Boulanger, Adrien; Mathieu, Pierre; Sert, Cagri; Sisto, Alessandro Large deviations for random walks on Gromov-hyperbolic spaces, Ann. Sci. Éc. Norm. Supér. (4), Volume 56 (2023) no. 3, pp. 885-944 | DOI | MR | Zbl

[8] Bridson, Martin R.; Haefliger, André Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, 319, Springer, 1999 | Zbl | DOI | MR

[9] Brown, Bruce M. Martingale Central Limit Theorems, Ann. Math. Stat., Volume 42 (1971) no. 1, pp. 59-66 | MR | Zbl | DOI

[10] Caprace, Pierre-Emmanuel; Fujiwara, Koji Rank-one isometries of buildings and quasi-morphisms of Kac–Moody groups, Geom. Funct. Anal., Volume 19 (2010) no. 5, pp. 1296-1319 | DOI | Zbl | MR

[11] Caprace, Pierre-Emmanuel; Sageev, Michah Rank rigidity for CAT(0) cube complexes, Geom. Funct. Anal., Volume 21 (2011) no. 4, pp. 851-891 | Zbl | DOI | MR

[12] Choi, Inhyeok Random walks and contracting elements I: Deviation inequality and Limit laws (2022) | arXiv | Zbl | DOI

[13] Croke, Christopher B.; Kleiner, Bruce Spaces with nonpositive curvature and their ideal boundaries, Topology, Volume 39 (2000) no. 3, pp. 549-556 | MR | DOI | Zbl

[14] Fernós, Talia; Lécureux, Jean; Mathéus, Frédéric Contact graphs, boundaries, and a central limit theorem for CAT(0) cubical complexes, Groups Geom. Dyn., Volume 18 (2024) no. 2, pp. 677-704 | Zbl | DOI | MR

[15] Furstenberg, Harry Noncommuting Random Products, Trans. Am. Math. Soc., Volume 108 (1963) no. 3, pp. 377-428 | DOI | MR | Zbl

[16] Furstenberg, Harry; Kesten, Harry Products of Random Matrices, Ann. Math. Stat., Volume 31 (1960) no. 2, pp. 457-469 | DOI | MR | Zbl

[17] Genevois, Anthony Hyperbolicities in CAT (0) cube complexes, Enseign. Math., Volume 65 (2019) no. 1-2, pp. 33-100 | Zbl | DOI | MR

[18] Gouëzel, Sébastien Exponential bounds for random walks on hyperbolic spaces without moment conditions, Tunis. J. Math., Volume 4 (2022) no. 4, pp. 635-671 | MR | Zbl | DOI

[19] Guivarc’h, Yves; Raugi, Albert Frontière de Furstenberg, propriétés de contraction et théorèmes de convergence, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 69 (1985) no. 2, pp. 187-242 | Zbl | DOI | MR

[20] Hagen, Mark F. Weak hyperbolicity of cube complexes and quasi-arboreal groups, J. Topol., Volume 7 (2014) no. 2, pp. 385-418 | DOI | MR | Zbl

[21] Hamenstädt, Ursula Rank-one isometries of proper CAT (0)-spaces, Discrete groups and geometric structures (Contemporary Mathematics), Volume 501, American Mathematical Society, 2009, pp. 43-59 | DOI | MR | Zbl

[22] Horbez, Camille Central limit theorems for mapping class groups and Out(F N ), Geom. Topol., Volume 22 (2018) no. 1, pp. 105-156 | MR | Zbl | DOI

[23] Izeki, Hiroyasu Isometric group actions with vanishing rate of escape on CAT (0) spaces, Geom. Funct. Anal., Volume 33 (2023) no. 1, pp. 170-244 | Zbl | DOI | MR

[24] Kaimanovich, Vadim A. The Poisson formula for groups with hyperbolic properties, Ann. Math. (2), Volume 152 (2000) no. 3, pp. 659-692 | Zbl | DOI | MR

[25] Karlsson, Anders; Margulis, Gregory A Multiplicative Ergodic Theorem and Nonpositively Curved Spaces, Commun. Math. Phys., Volume 208 (1999), pp. 107-123 | Zbl | MR | DOI

[26] Le Bars, Corentin Random walks and rank one isometries on CAT(0) spaces (2022) | arXiv | Zbl | DOI

[27] Le Bars, Corentin Marches aléatoires et éléments contractants sur des espaces CAT(0), Ph. D. Thesis, Université Paris-Saclay (2023) (directed by Jean Lécureux, http://www.theses.fr/2023upasm017)

[28] Le Page, Emile Théorèmes limites pour les produits de matrices aléatoires, Probability Measures on Groups (Heyer, Herbert, ed.), Springer (1982), pp. 258-303 | DOI | Zbl

[29] Maher, Joseph; Tiozzo, Giulio Random walks on weakly hyperbolic groups, J. Reine Angew. Math., Volume 742 (2018), pp. 187-239 | DOI | MR | Zbl

[30] Mathieu, Pierre; Sisto, Alessandro Deviation inequalities for random walks, Duke Math. J., Volume 169 (2020) no. 5, pp. 961-1036 | MR | DOI | Zbl

[31] Petyt, Harry; Spriano, Davide; Zalloum, Abdul Hyperbolic models for CAT (0) spaces, Adv. Math., Volume 450 (2024), 109742, 66 pages | Zbl | DOI | MR

[32] Qing, Yulan; Rafi, Kasra Sublinearly Morse boundary I: CAT(0) spaces, Adv. Math., Volume 404 (2022), 108442, 51 pages | DOI | MR | Zbl

[33] Qing, Yulan; Rafi, Kasra; Tiozzo, Giulio Sublinearly Morse Boundary II: Proper geodesic spaces, Geom. Topol., Volume 28 (2024) no. 4, pp. 1829-1889 | DOI | MR | Zbl

Cited by Sources: