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Renewal theory for random walks on surface groups

Published online by Cambridge University Press:  12 May 2016

PETER HAÏSSINSKY ,
PIERRE MATHIEU  and
SEBASTIAN MÜLLER
PETER HAÏSSINSKY
Affiliation:
IMT, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 9, France email [email protected]
PIERRE MATHIEU
Affiliation:
Aix Marseille Université, CNRS, Centrale Marseille, I2M UMR 7373, 13453, Marseille, France email [email protected], [email protected]
SEBASTIAN MÜLLER
Affiliation:
Aix Marseille Université, CNRS, Centrale Marseille, I2M UMR 7373, 13453, Marseille, France email [email protected], [email protected]
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Abstract

We construct a renewal structure for random walks on surface groups. The renewal times are defined as times when the random walks enter a particular type of cone and never leave it again. As a consequence, the trajectory of the random walk can be expressed as an aligned union of independent and identically distributed trajectories between the renewal times. Once having established this renewal structure, we prove a central limit theorem for the distance to the origin under exponential moment conditions. Analyticity of the speed and of the asymptotic variance are natural consequences of our approach. Furthermore, our method applies to groups with infinitely many ends and therefore generalizes classic results on central limit theorems on free groups.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 1 , February 2018 , pp. 155 - 179
Copyright
© Cambridge University Press, 2016 

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References

Bellman, R.. Limit theorems for non-commutative operations. I. Duke Math. J. 21 (1954), 491–500.Google Scholar
Benoist, Y. and Quint, J.-.F. Central limit theorem for hyperbolic groups. Izv. Math., to appear. Available at: http://www.math.u-psud.fr/∼benoist/prepubli/14centralhyperbolic.pdf.Google Scholar
Berg, C. and Christensen, J. P. R.. On the relation between amenability of locally compact groups and the norms of convolution operators. Math. Ann. 208 (1974), 149–153.Google Scholar
Billingsley, P.. Convergence of probability measures. Wiley Series in Probability and Statistics: Probability and Statistics, 2nd edn. Wiley, New York, 1999.Google Scholar
Björklund, M.. Central limit theorems for Gromov hyperbolic groups. J. Theoret. Probab. 23(3) (2010), 871–887.Google Scholar
Calegari, D.. The ergodic theory of hyperbolic groups. Geometry and Topology Down Under (Contemporary Mathematics, 597) . American Mathematical Society, Providence, RI, 2013, pp. 15–52.Google Scholar
Coornaert, M. and Papadopoulos, A.. Symbolic Dynamics and Hyperbolic Groups (Lecture Notes in Mathematics, 1539) . Springer, Berlin, 1993.Google Scholar
Derriennic, Y. and Guivarc’h, Y.. Théorème de renouvellement pour les groupes non moyennables. C. R. Acad. Sci. Paris Sér. A-B 277 (1973), A613–A615.Google Scholar
Dyubina, A. G.. An example of the rate of departure to infinity for a random walk on a group. Uspekhi Mat. Nauk 54(5(329)) (1999), 159–160.Google Scholar
Epstein, D. B. A., Cannon, J. W., Holt, D. F., Levy, V. F., Paterson, M. S. and Thurston, W. P.. Word Processing in Groups. Jones and Bartlett Publishers, Boston, MA, 1992.Google Scholar
Èrshler, A.. Asymptotics of drift and entropy for a random walk on groups. Uspekhi Mat. Nauk 56(3(339)) (2001), 179–180.Google Scholar
Furstenberg, H. and Kesten, H.. Products of random matrices. Ann. Math. Stat. 31 (1960), 457–469.Google Scholar
Gilch, L. A.. Rate of escape of random walks on free products. J. Aust. Math. Soc. 83(1) (2007), 31–54.Google Scholar
Gilch, L. A.. Rate of escape of random walks on regular languages and free products by amalgamation of finite groups. Fifth Colloquium on Mathematics and Computer Science (Discrete Mathematics & Theoretical Computer Science Proceedings) . Association of Discrete Mathematics and Theoretical Computer Science, Nancy, 2008, pp. 405–420.Google Scholar
Gilch, L. and Ledrappier, F.. Regularity of the drift and entropy of random walks on groups. Publ. Mat. Urug. 14 (2013), 147–158.Google Scholar
Gouëzel, S.. Analyticity of the entropy and the escape rate of random walks in hyperbolic groups, Preprint, 2015, http://arxiv.org/abs/1509.06859.Google Scholar
Gouëzel, S. and Lalley, S. P.. Random walks on co-compact Fuchsian groups. Ann. Sci. Éc. Norm. Supér. (4) 46(1) (2013), 129–173.Google Scholar
Guivarc’h, Y.. Sur la loi des grands nombres et le rayon spectral d’une marche aléatoire. Conference on Random Walks (Kleebach, 1979) (French) (Astérisque, 74(3)) . Société Mathématique de France, Paris, 1980, pp. 47–98.Google Scholar
Guivarc’h, Y. and Le Page, É.. Simplicité de spectres de Lyapunov et propriété d’isolation spectrale pour une famille d’opérateurs de transfert sur l’espace projectif. de Gruyter, Berlin, 2004, pp. 181–259.Google Scholar
Kaimanovich, V. A.. The Poisson formula for groups with hyperbolic properties. Ann. of Math. (2) 152(3) (2000), 659–692.Google Scholar
Kaimanovich, V. A. and Èrshler, A. G.. Continuity of asymptotic characteristics for random walks on hyperbolic groups. Funktsional. Anal. i Prilozhen. 47(2) (2013), 84–89.Google Scholar
Lalley, S. P.. Finite range random walk on free groups and homogeneous trees. Ann. Probab. 21(4) (1993), 2087–2130.Google Scholar
Le Page, É.. Théorèmes de la limite centrale pour certains produits de matrices aléatoires. C. R. Acad. Sci. Paris Sér. I Math. 292(6) (1981), 379–382.Google Scholar
Ledrappier, F.. Some asymptotic properties of random walks on free groups. CRM Proc. Lectures Notes 28 (2001).Google Scholar
Ledrappier, F.. Analyticity of the entropy for some random walks. Groups Geom. Dyn. 6(2) (2012), 317–333.Google Scholar
Ledrappier, F.. Regularity of the entropy for random walks on hyperbolic groups. Ann. Probab. 41(5) (2013), 3582–3605.Google Scholar
Mairesse, J. and Mathéus, F.. Random walks on free products of cyclic groups. J. Lond. Math. Soc. (2) 75(1) (2007), 47–66.Google Scholar
Mairesse, J. and Mathéus, F.. Randomly growing braid on three strands and the manta ray. Ann. Appl. Probab. 17(2) (2007), 502–536.Google Scholar
Nagnibeda, T. and Woess, W.. Random walks on trees with finitely many cone types. J. Theoret. Probab. 15(2) (2002), 383–422.Google Scholar
Pollicott, M. and Sharp, R.. Statistics of matrix products in hyperbolic geometry. International Conference on Dynamical Numbers: Interplay Between Dynamical Systems and Number Theory. Vol. 532. American Mathematical Society, Providence, RI, 2010, pp. 213–230.Google Scholar
Sawyer, S. and Steger, T.. The rate of escape for anisotropic random walks in a tree. Probab. Theory Related Fields 76 (1987), 207–230.Google Scholar
Series, C.. The infinite word problem and limit sets in Fuchsian groups. Ergod. Th. & Dynam. Sys. 1(3) (1982), 337–360.Google Scholar
Tutubalin, V. N.. Limit theorems for a product of random matrices. Teor. Verojatn. Primen. 10 (1965), 19–32.Google Scholar
Tutubalin, V. N.. On random walk in the Lobachevsky plane. Theory Probab. Appl. 13 (1968), 487–490.CrossRefGoogle Scholar
Zeitouni, O.. Random walks in random environment. Lectures on Probability Theory and Statistics (Lecture Notes in Mathematics, 1837) . Springer, Berlin, 2004, pp. 189–312.Google Scholar