Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

Local limit theorem for symmetric random walks in Gromov-hyperbolic groups


Author: Sébastien Gouëzel
Journal: J. Amer. Math. Soc. 27 (2014), 893-928
MSC (2010): Primary 05C81, 60J50, 20F67
DOI: https://doi.org/10.1090/S0894-0347-2014-00788-8
Published electronically: March 20, 2014
MathSciNet review: 3194496
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Completing a strategy of Gouëzel and Lalley, we prove a local limit theorem for the random walk generated by any symmetric finitely supported probability measure on a non-elementary Gromov-hyperbolic group: denoting by $ R$ the inverse of the spectral radius of the random walk, the probability to return to the identity at time $ n$ behaves like $ C R^{-n}n^{-3/2}$. An important step in the proof is to extend Ancona's results on the Martin boundary up to the spectral radius: we show that the Martin boundary for $ R$-harmonic functions coincides with the geometric boundary of the group. In Appendix A, we explain how the symmetry assumption of the measure can be dispensed with for surface groups.


References [Enhancements On Off] (What's this?)

  • [Ale02] Georgios K. Alexopoulos, Random walks on discrete groups of polynomial volume growth, Ann. Probab. 30 (2002), no. 2, 723-801. MR 1905856 (2003d:60010), https://doi.org/10.1214/aop/1023481007
  • [Anc87] Alano Ancona, Negatively curved manifolds, elliptic operators, and the Martin boundary, Ann. of Math. (2) 125 (1987), no. 3, 495-536. MR 890161 (88k:58160), https://doi.org/10.2307/1971409
  • [BHM11] Sébastien Blachère, Peter Haïssinsky, and Pierre Mathieu, Harmonic measures versus quasiconformal measures for hyperbolic groups, Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), 683-721. MR 2919980
  • [Bou81] Philippe Bougerol, Théorème central limite local sur certains groupes de Lie, Ann. Sci. Éc. Norm. Supér. (4) 14 (1981), no. 4, 403-432 (1982) (French). MR 654204 (83g:60019)
  • [BS00] Mario Bonk and Oded Schramm, Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal. 10 (2000), 266-306 (French). MR 1771428
  • [Can84] James W. Cannon, The combinatorial structure of cocompact discrete hyperbolic groups, Geom. Dedicata 16 (1984), no. 2, 123-148. MR 758901 (86j:20032), https://doi.org/10.1007/BF00146825
  • [CF10] Danny Calegari and Koji Fujiwara, Combable functions, quasimorphisms, and the central limit theorem, Ergodic Theory Dynam. Systems 30 (2010), no. 5, 1343-1369. MR 2718897 (2011k:20088), https://doi.org/10.1017/S0143385709000662
  • [CNW73] J. Chover, P. Ney, and S. Wainger, Functions of probability measures, J. Anal. Math. 26 (1973), 255-302. MR 0348393 (50 #891)
  • [Coo93] Michel Coornaert, Mesures de Patterson-Sullivan sur le bord d'un espace hyperbolique au sens de Gromov, Pacific J. Math. 159 (1993), no. 2, 241-270 (French, with French summary). MR 1214072 (94m:57075)
  • [DPPS11] Françoise Dal'Bo, Marc Peigné, Jean-Claude Picaud, and Andrea Sambusetti, On the growth of quotients of Kleinian groups, Ergodic Theory Dynam. Systems 31 (2011), no. 3, 835-851. MR 2794950 (2012f:37069), https://doi.org/10.1017/S0143385710000131
  • [Fur02] Alex Furman, Random walks on groups and random transformations, Handbook of dynamical systems, Vol. 1A, North-Holland, Amsterdam, 2002, pp. 931-1014. MR 1928529 (2003j:60065), https://doi.org/10.1016/S1874-575X(02)80014-5
  • [GdlH90] Étienne Ghys and Pierre de la Harpe, Sur les groupes hyperboliques d'après Mikhael Gromov, Progress in Mathematics, Vol. 83, Birkhäuser Boston Inc., Boston, MA, 1988. Papers from the Swiss Seminar on Hyperbolic Groups held in Bern. MR 1086648
  • [GL13] Sébastien Gouëzel and Steven P. Lalley, Random walks on co-compact Fuchsian groups, Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), no. 1, 129-173 (2013) (English, with English and French summaries). MR 3087391
  • [INO08] Masaki Izumi, Sergey Neshveyev, and Rui Okayasu, The ratio set of the harmonic measure of a random walk on a hyperbolic group, Israel J. Math. 163 (2008), 285-316. MR 2391133 (2009f:60094), https://doi.org/10.1007/s11856-008-0013-6
  • [Kat66] Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473 (34 #3324)
  • [Kre85] Ulrich Krengel, Ergodic theorems, de Gruyter Studies in Mathematics, vol. 6, Walter de Gruyter & Co., Berlin, 1985. With a supplement by Antoine Brunel. MR 797411 (87i:28001)
  • [Lal93] Steven P. Lalley, Finite range random walk on free groups and homogeneous trees, Ann. Probab. 21 (1993), no. 4, 2087-2130. MR 1245302 (94m:60051)
  • [Led13] François Ledrappier, Regularity of the entropy for random walks on hyperbolic groups, Ann. Probab. 41 (2013), no. 5, 3582-3605. MR 3127892, https://doi.org/10.1214/12-AOP748
  • [PP90] William Parry and Mark Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque 187-188 (1990), 268 (English, with French summary). MR 1085356 (92f:58141)
  • [Ser83] Caroline Series, Martin boundaries of random walks on Fuchsian groups, Israel J. Math. 44 (1983), no. 3, 221-242. MR 693661 (85b:60068), https://doi.org/10.1007/BF02760973
  • [Woe00] Wolfgang Woess, Random walks on infinite graphs and groups, Cambridge Tracts in Mathematics, vol. 138, Cambridge University Press, Cambridge, 2000. MR 1743100 (2001k:60006)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 05C81, 60J50, 20F67

Retrieve articles in all journals with MSC (2010): 05C81, 60J50, 20F67


Additional Information

Sébastien Gouëzel
Affiliation: IRMAR, CNRS UMR 6625, Université de Rennes 1, 35042 Rennes, France
Email: sebastien.gouezel@univ-rennes1.fr

DOI: https://doi.org/10.1090/S0894-0347-2014-00788-8
Keywords: Local limit theorem, random walk, hyperbolic group, spectral radius, Martin boundary, transfer operator
Received by editor(s): September 17, 2012
Received by editor(s) in revised form: September 17, 2013
Published electronically: March 20, 2014
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society