Random walks on co-compact fuchsian groups

Gouëzel, Sébastien; Lalley, Steven P.

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 46 (2013) no. 1, pp. 131-175.

Considérons une marche aléatoire symétrique à support fini sur un groupe fuchsien co-compact. Nous montrons que la fonction de Green à son rayon de convergence $R$ décroît exponentiellement vite en fonction de la distance à l’origine. Nous montrons également que les inégalités d’Ancona s’étendent jusqu’au paramètre $R$, et par conséquent que la frontière de Martin pour les $R$-potentiels s’identifie avec la frontière géométrique $S^1$. De plus, le noyau de Martin correspondant est höldérien. Ces résultats sont utilisés pour démontrer un théorème limite local pour les probabilités de transition : dans le cas apériodique, $p^n(x,y)\sim C_{x,y}{R}^{-n}{n}^{-3/2}$.

It is proved that the Green’s function of a symmetric finite range random walk on a co-compact Fuchsian group decays exponentially in distance at the radius of convergence $R$. It is also shown that Ancona’s inequalities extend to $R$, and therefore that the Martin boundary for $R$-potentials coincides with the natural geometric boundary $S^1$, and that the Martin kernel is uniformly Hölder continuous. Finally, this implies a local limit theorem for the transition probabilities: in the aperiodic case, $p^n(x,y)\sim C_{x,y}{R}^{-n}{n}^{-3/2}$.

[1] A. Ancona, Negatively curved manifolds, elliptic operators, and the Martin boundary, Ann. of Math. 125 (1987), 495-536. | MR

[2] A. Ancona, Positive harmonic functions and hyperbolicity, in Potential theory-surveys and problems (Prague, 1987), Lecture Notes in Math. 1344, Springer, 1988, 1-23. | MR

[3] M. T. Anderson & R. Schoen, Positive harmonic functions on complete manifolds of negative curvature, Ann. of Math. 121 (1985), 429-461. | MR

[4] K. B. Athreya & P. E. Ney, Branching processes, Grundl. der math. Wiss. 196, Springer, 1972. | MR

[5] V. Baladi, Positive transfer operators and decay of correlations, Advanced Series in Nonlinear Dynamics 16, World Scientific Publishing Co. Inc., 2000. | MR

[6] N. H. Bingham, C. M. Goldie & J. L. Teugels, Regular variation, Encyclopedia of Math. and its Appl. 27, Cambridge Univ. Press, 1987. | MR

[7] S. Blachère & S. Brofferio, Internal diffusion limited aggregation on discrete groups having exponential growth, Probab. Theory Related Fields 137 (2007), 323-343. | MR

[8] S. Blachère, P. Haïssinsky & P. Mathieu, Asymptotic entropy and Green speed for random walks on countable groups, Ann. Probab. 36 (2008), 1134-1152. | MR

[9] S. Blachère, P. Haïssinsky & P. Mathieu, Harmonic measures versus quasiconformal measures for hyperbolic groups, Ann. Sci. Éc. Norm. Supér. 44 (2011), 683-721. | Numdam | MR

[10] P. Bougerol, Théorème central limite local sur certains groupes de Lie, Ann. Sci. École Norm. Sup. 14 (1981), 403-432. | MR

[11] R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Math. 470, Springer, 1975. | MR

[12] J. W. Cannon, The combinatorial structure of cocompact discrete hyperbolic groups, Geom. Dedicata 16 (1984), 123-148. | MR

[13] J. W. Cannon, Almost convex groups, Geom. Dedicata 22 (1987), 197-210. | MR

[14] J. W. Cannon, The theory of negatively curved spaces and groups, in Ergodic theory, symbolic dynamics, and hyperbolic spaces (Trieste, 1989), Oxford Sci. Publ., Oxford Univ. Press, 1991, 315-369. | MR

[15] D. I. Cartwright, Singularities of the Green function of a random walk on a discrete group, Monatsh. Math. 113 (1992), 183-188. | MR

[16] I. Chatterji & K. Ruane, Some geometric groups with rapid decay, Geom. Funct. Anal. 15 (2005), 311-339. | MR

[17] W. Feller, An introduction to probability theory and its applications. Vol. II., Second edition, John Wiley & Sons Inc., 1971. | MR

[18] P. Gerl & W. Woess, Local limits and harmonic functions for nonisotropic random walks on free groups, Probab. Theory Relat. Fields 71 (1986), 341-355. | MR

[19] É. Ghys & P. De La Harpe, Espaces métriques hyperboliques, in Sur les groupes hyperboliques d'après Mikhael Gromov (Bern, 1988), Progr. Math. 83, Birkhäuser, 1990, 27-45. | MR

[20] M. Gromov, Hyperbolic groups, in Essays in group theory, Math. Sci. Res. Inst. Publ. 8, Springer, 1987, 75-263. | MR

[21] U. Hamenstädt, Harmonic measures, Hausdorff measures and positive eigenfunctions, J. Differential Geom. 44 (1996), 1-31. | MR

[22] M. Izumi, S. Neshveyev & R. Okayasu, The ratio set of the harmonic measure of a random walk on a hyperbolic group, Israel J. Math. 163 (2008), 285-316. | MR

[23] I. Kapovich & N. Benakli, Boundaries of hyperbolic groups, in Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), Contemp. Math. 296, Amer. Math. Soc., 2002, 39-93. | MR

[24] S. Katok, Fuchsian groups, Chicago Lectures in Mathematics, University of Chicago Press, 1992. | MR

[25] H. Kesten, Full Banach mean values on countable groups, Math. Scand. 7 (1959), 146-156. | MR

[26] S. P. Lalley, Renewal theorems in symbolic dynamics, with applications to geodesic flows, non-Euclidean tessellations and their fractal limits, Acta Math. 163 (1989), 1-55. | MR

[27] S. P. Lalley, Finite range random walk on free groups and homogeneous trees, Ann. Probab. 21 (1993), 2087-2130. | MR

[28] S. P. Lalley, Random walks on regular languages and algebraic systems of generating functions, in Algebraic methods in statistics and probability (Notre Dame, IN, 2000), Contemp. Math. 287, Amer. Math. Soc., 2001, 201-230. | MR

[29] F. Ledrappier, A renewal theorem for the distance in negative curvature, in Stochastic analysis (Ithaca, NY, 1993), Proc. Sympos. Pure Math. 57, Amer. Math. Soc., 1995, 351-360. | MR

[30] F. Ledrappier, Some asymptotic properties of random walks on free groups, in Topics in probability and Lie groups: boundary theory, CRM Proc. Lecture Notes 28, Amer. Math. Soc., 2001, 117-152. | MR

[31] T. Nagnibeda & W. Woess, Random walks on trees with finitely many cone types, J. Theoret. Probab. 15 (2002), 383-422. | MR

[32] W. Parry & M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque 187-188 (1990). | MR

[33] M. Picardello & W. Woess, Examples of stable Martin boundaries of Markov chains, in Potential theory (Nagoya, 1990), de Gruyter, 1992, 261-270. | MR

[34] C. Series, The infinite word problem and limit sets in Fuchsian groups, Ergodic Theory Dynam. Systems 1 (1981), 337-360. | MR

[35] C. Series, Martin boundaries of random walks on Fuchsian groups, Israel J. Math. 44 (1983), 221-242. | MR

[36] H. S. Wall, Analytic theory of continued fractions, D. Van Nostrand Company, Inc., New York, N. Y., 1948. | MR

[37] W. Woess, A local limit theorem for random walks on certain discrete groups, in Probability measures on groups (Oberwolfach, 1981), Lecture Notes in Math. 928, Springer, 1982, 467-477. | MR

[38] W. Woess, Nearest neighbour random walks on free products of discrete groups, Boll. Un. Mat. Ital. B 5 (1986), 961-982. | MR

[39] W. Woess, Random walks on infinite graphs and groups, Cambridge Tracts in Mathematics 138, Cambridge Univ. Press, 2000. | MR

[40] W. Woess, Context-free pairs of groups II-cuts, tree sets, and random walks, Discrete Math. 312 (2012), 157-173. | MR