Central limit theorem on CAT(0) spaces with contracting isometries
[Théorème de la limite centrale pour les marches aléatoires sur des espaces CAT(0) en présence d’éléments contractants]
Le Bars, Corentin1
1 Weizmann Institute of Science Rehovot (Israel)
Annales de l'Institut Fourier, Online first, 42 p.

Let G be a group acting on a 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 CAT(0) spaces. As a side result, we prove that the probability that the nth-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 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 CAT(0) introduits par H. Petyt, A. Zalloum et D. Spriano, une construction analogue au graphe de contact pour les complexes cubiques 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 +∞.

Reçu le :
Révisé le :
Accepté le :
Première publication :
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@unpublished{AIF_0__0_0_A5_0,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},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2025},
     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
%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) | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

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

[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 | DOI | MR | Zbl

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

[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 | MR | Zbl

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

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

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

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

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

[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 | DOI | MR | Zbl

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

[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

Cité par Sources :