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Context-free languages and random walks on groups. (English) Zbl 0637.60014
The Green function of an arbitrary, finitely supported random walk on a discrete group with context-free word problem is algebraic. It is shown how this theorem can be deduced from basic results of formal language theory. Context-free groups are precisely the finite extensions of free groups.

60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60G50 Sums of independent random variables; random walks
03D40 Word problems, etc. in computability and recursion theory
68Q45 Formal languages and automata
Full Text: DOI
[1] Aomoto, K., Spectral theory on a free group and algebraic curves, J. fac. sci. univ. Tokyo, sec. IA, 31, 297-317, (1984) · Zbl 0583.60068
[2] Bender, E.A., Asymptotic methods in enumeration, SIAM review, 16, 485-515, (1974) · Zbl 0294.05002
[3] Cartwright, D.I.; Soardi, P.M., Random walks on free products, quotients and amalgams, Nagoya math. J., 102, 163-180, (1986) · Zbl 0592.60052
[4] Chomsky, N.; Schützenberger, M.P., The algebraic theory of context-free languages, (), 118-161 · Zbl 0148.00804
[5] Dunwoody, M.J., The accessibility of finitely presented groups, Inventiones math., 81, 449-457, (1985) · Zbl 0572.20025
[6] Gerl, P., Wahrscheinlichkeitsmaβe auf disktreten gruppen, Archiv math., 31, 611-619, (1978)
[7] Gerl, P., Ein konvergenzsatz für faltungspotenzen, (), 120-125 · Zbl 0405.60011
[8] Gerl, P.; Woess, W., Local limits and harmonic functions for nonistropic walks on free groups, Probab. th. rel. fields, 71, 341-355, (1986) · Zbl 0562.60011
[9] Guivarc’h, Y.; Keane, M.; Roynette, B., Marches aléatoires sur LES groupes de Lie, () · Zbl 0367.60081
[10] Harrison, M.A., Introduction to formal language theory, (1978), Addison-Wesley London · Zbl 0411.68058
[11] Kaimanovich, V.A.; Vershik, A.M., Random walks on discrete groups: boundary and entropy, Ann. probab., 11, 457-490, (1983) · Zbl 0641.60009
[12] Kesten, H., Symmetric random walks on groups, Trans. amer. math. soc., 92, 336-354, (1959) · Zbl 0092.33503
[13] Kuich, W.; Salomaa, A., Semerings, automata, languages, (1985), Springer Berlin
[14] Muller, D.E.; Schupp, P.E., Groups, the theory of ends, and context-free languages, J. comp. syst. sci., 26, 295-310, (1983) · Zbl 0537.20011
[15] Picardello, M.A., Spherical functions and local limit theorems on free groups, Ann. math. pur. appl., 33, 177-191, (1983) · Zbl 0527.60011
[16] Sawyer, S., Isotropic random walks in a tree, Z. wahrsch. verw. gebiete, 42, 279-292, (1978) · Zbl 0362.60075
[17] Spitzer, F., Principles of random walk, (1964), Van Nostrand Princeton · Zbl 0119.34304
[18] Stallings, J., Group theory and three-dimensional manifolds, (1971), Yale Univ. Press New Haven · Zbl 0241.57001
[19] Steger, T., Harmonic analysis for an anisotropic random walk on a homogeneous tree, ()
[20] van der Waerden, B.L., Einführung in die algebraische geometrie, (1939), Springer Berlin · Zbl 0021.25001
[21] Woess, W., Nearest neighbour random walks on free products of discrete groups, Boll. un. mat. it., 5-B, 961-982, (1986) · Zbl 0627.60012
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