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Random walks on free products, quotients and amalgams

Published online by Cambridge University Press:  22 January 2016

Donald I. Cartwright  and
P. M. Soardi
Donald I. Cartwright
Affiliation:
Department of Pure Mathematics, The University of Sydney, Sydney N.S.W. 2006, Australia
P. M. Soardi
Affiliation:
Dipartimento di Matematica, Università di Milano, Via C. Saldini 50, Milano, 20133, Italy
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Suppose that G is a discrete group and p is a probability measure on G. Consider the associated random walk {Xn} on G. That is, let Xn = Y1Y2 … Yn, where the Yj’s are independent and identically distributed G-valued variables with density p. An important problem in the study of this random walk is the evaluation of the resolvent (or Green’s function) R(z, x) of p. For example, the resolvent provides, in principle, the values of the n step transition probabilities of the process, and in several cases knowledge of R(z, x) permits a description of the asymptotic behaviour of these probabilities.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 102 , June 1986 , pp. 163 - 180
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1986

References

[ 1 ] Aomoto, K., Spectral theory on a free group and algebraic curves, J. Fac. Sci. Univ. Tokyo Sec. 1A, 31 (1984), 297–317.Google Scholar
[ 2 ] Aomoto, K. and Kato, Y., Green functions and spectrums in a free product of cyclic groups, preprint.Google Scholar
[ 3 ] Bender, E. A., Asymptotic methods in enumeration, SIAM Review, 16 (1974), 485–515.Google Scholar
[ 4 ] Cartier, P., Fonctions harmoniques sur un arbre, Symposia Math., 9 (1972), 203270.Google Scholar
[ 5 ] Cartwright, D. I. and Soardi, P. M., Harmonie analysis on the free product of two cyclic groups, J. Funct. Anal., 65 (1986), 147–171.CrossRefGoogle Scholar
[ 6 ] Chihara, T. S., An introduction to orthogonal polynomials, Gordon and Breach, New York-London-Paris, 1978.Google Scholar
[ 7 ] Cohen, J. M. and Trenholme, A. R., Orthogonal polynomials with a constant recursion formula and an application to harmonic analysis, J. Funct. Anal., 59 (1984), 175–184.Google Scholar
[ 8 ] Faraut, J. and Picardello, M. A., The Plancherel measure for symmetric graphs, Ann. Mat. Pura Appl., 138 (1984), 151–155.Google Scholar
[ 9 ] Figà-Talamanca, A. and Picardello, M. A., Harmonic analysis on free groups, Lecture Notes in Pure and Applied Mathematics, 87, Marcel Dekker, New York, 1983.Google Scholar
[10] Geri, P., Ein Gleichverteilungssatz auf F2 , Probability Measures on Groups, Lecture Notes in Math., 706, Springer-Verlag, Berlin-Heidelberg-New York, 1979, pp. 126–130.Google Scholar
[11] Geri, P., A local central limit theorem on some groups, The First Pannonian Sym posium on Mathematical Statistics, Lecture Notes in Statistics 8, Springer-Verlag, Berlin-Heidelberg-New York, 1981, pp. 73–82.Google Scholar
[12] P. Geri and Woess, W., Local limits and harmonic functions for non-isotropic random walks on free groups, Probab. Th. Rei. Fields, 71 (1986), 341–355.Google Scholar
[13] Kesten, H., Symmetric walks on groups, Trans. Amer. Math. Soc., 92 (1959), 336–354.CrossRefGoogle Scholar
[14] Kuhn, G. and Soardi, P. M., The Plancherel measure for polygonal graphs, Ann. Mat. Pura Appl., 134 (1983), 393–401.Google Scholar
[15] Picardello, M. A., Spherical functions and local limit theorems on free groups, Ann. Mat. Pura Appl., 133 (1983), 177–191.Google Scholar
[16] Picardello, M. A. and Woess, W., Random walks on amalgams, Monatsh. Math., 100 (1985), 21–33.CrossRefGoogle Scholar
[17] Sawyer, S., Isotropic random walks in a tree, Z. Wahrsch. Verw. Gebiete, 42 (1978), 279–292.Google Scholar
[18] Schaefer, H. H., Banach lattices and positive operators, Grundlehren der Math. Wissen. 215, Springer-Verlag, Berlin-Heidelberg-New York, 1974.CrossRefGoogle Scholar
[19] Serre, J. P., Trees, Springer-Verlag, Berlin-Heidelberg-New York, 1980.CrossRefGoogle Scholar
[20] Soardi, P. M., The resolvent for simple random walks on the free product of two discrete groups, preprint.Google Scholar
[21] Szegö, G., Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publications, Vol. XXIII, fourth edition, Amer. Math. Soc, Providence Rhode Island, 1975.Google Scholar
[22] Woess, W., A random walk on the free product of finite groups, in Springer Lecture Notes in Mathematics 1064, (1984), 467–470.Google Scholar