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Rate of escape of random walks on free products

Part of: Structure and classification of infinite or finite groups Probability theory on algebraic and topological structures Stochastic processes

Published online by Cambridge University Press:  09 April 2009

Lorenz A. Gilch
Lorenz A. Gilch
Affiliation:
University of Technology GrazInstitut fur Mathematische Strukturtheorie (Math. C)Steyrergasse 30 [email protected]
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Abstract

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Suppose we are given the free product V of a finite family of finite or countable sets (Vi)i∈∮ and probability measures on each Vi, which govern random walks on it. We consider a transient random walk on the free product arising naturally from the random walks on the Vi. We prove the existence of the rate of escape with respect to the block length, that is, the speed at which the random walk escapes to infinity, and furthermore we compute formulae for it. For this purpose, we present three different techniques providing three different, equivalent formulae.

Keywords

random walksfree productsrate of escape

MSC classification

Secondary: 60G50: Sums of independent random variables; random walks 20E06: Free products, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 83 , Issue 1 , August 2007 , pp. 31 - 54
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Cartwright, D. I., Kaimanovich, V. A. and Woess, W., ‘Random walks on the affine group of local fields and of homogeneous trees’, Ann. Inst. Fourier (Grenoble) 44 (1994), 1243–1288.CrossRefGoogle Scholar
[2]Cartwright, D. I. and Soardi, P. M., ‘Random walks on free products, quotients, and amalgams’, Nagoya Math. J. 102 (1986), 163–180.CrossRefGoogle Scholar
[3]Derriennic, Y., ‘Quelques applications du théorème ergodique sous-additif’, Astérisque 74 (1980), 183–201.Google Scholar
[4]Dyubina, A., ‘Characteristics of random walks on wreath products of groups’, J. Math. Sci. (5) 107 (2001), 4166–4171.CrossRefGoogle Scholar
[5]Erschler, A., ‘On the asymptotics of drift’, J. Math. Sci. (3) 121 (2004), 2437–2440.CrossRefGoogle Scholar
[6]Furstenberg, H., ‘Non commuting random products’, Trans. Amer. Math. Soc. 108 (1963), 377–128.CrossRefGoogle Scholar
[7]Guivarc'h, Y., ‘Sur la loi des grands nombres et le rayon spectral d'une marche aléatoire’, Astérisque 74 (1980), 47–98.Google Scholar
[8]Kaimanovich, V. A. and Vershik, A. M., ‘Random walks on discrete groups: boundary and entropy’, Ann. Probab. 11 (1983), 457–490.CrossRefGoogle Scholar
[9]Kingman, J. F. C., ‘The ergodic theory of subadditive processes’, J. Roy. Statist. Soc., Ser. B 30 (1968), 499–510.Google Scholar
[10]Ledrappier, F., Some Asymptotic Properties of Random Walks on Free Groups CRM Proceedings and Lecture Notes 28 (Amer. Math. Soc, Providence, RI, 2001) pp. 117–152.CrossRefGoogle Scholar
[11]Lyons, R., Pemantle, R. and Peres, Y., ‘Random walks on the lamplighter group’, Ann. Probab. (4) 24 (1996), 1993–2006.CrossRefGoogle Scholar
[12]Mairesse, J., ‘Random walks on groups and monoids with a markovian harmonic measure’, Technical Report Research Report LIAFA 2004–2005, (Univ. Paris 7, 2004).Google Scholar
[13]Mairesse, J., ‘Randomly growing braid on three strands and the manta ray’, Technical Report Report LIAFA 2005–2001, (Univ. Paris 7, 2005).Google Scholar
[14]Mairesse, J. and Matheus, F., ‘Random walks on free products of cyclic groups and on Artin groups with two generators’, Technical Report Research Report LIAFA 2004–006, (Univ. Paris 7, 2004).Google Scholar
[15]McLaughlin, J. C., Random walks and convolution operators onfree products (Ph.D. Thesis, New York Univ., 1986).Google Scholar
[16]Nagnibeda, T. and Woess, W., ‘Random walks on trees with finitely many cone types’, J. Theoret. Probab. 15 (2002), 399–438.CrossRefGoogle Scholar
[17]Sawyer, S. and Steger, T., ‘The rate of escape for anisotropic random walks in a tree’, Probab. Theory Related Fields 76 (1987), 207–230.CrossRefGoogle Scholar
[18]Soardi, P. M., ‘Simple random walks on.2 * /2’, Symposia Math. 29 (1986), 303–309.Google Scholar
[19]Varopoulos, N. Th., ‘Long range estimates for Markov chains’, Bull. Sci. Math. (2) 109 (1985), 225–252.Google Scholar
[20]Voiculescu, D., ‘Addition of certain non-commuting random variables’, J. Funct. Anal. 66 (1986), 323–346.CrossRefGoogle Scholar
[21]Woess, W., ‘A description of the Martin boundary for nearest neighbour random walks on free products’, Probability Measures on Groups VII (1985), 203–215.Google Scholar
[22]Woess, W., ‘Nearest neighbour random walks on free products of discrete groups’, Boll. Un. Mat. Ital. (6)5-B (1986), 961–982.Google Scholar
[23]Woess, W., Random Walks on Infinite Graphs and Groups (Cambridge University Press, Cambridge, 2000).CrossRefGoogle Scholar