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Anna Erschler
Critical constants for recurrence of random walks on $G$-spaces
(Constantes critiques pour la récurrence des marches aléatoires sur les $G$-espace.)
Annales de l'institut Fourier, 55 no. 2 (2005), p. 493-509, doi: 10.5802/aif.2105
Article PDF | Reviews MR 2147898 | Zbl 02171516 | 1 citation in Cedram
Class. Math.: 20F65, 20E08, 60B15
Keywords: growth of groups, Grigorchuk groups, branch groups, random walks, recurrence, drift
We introduce the notion of a critical constant $c_{rt}$ for recurrence of random walks on
$G$-spaces. For a subgroup $H$ of a finitely generated group $G$ the critical constant is
an asymptotic invariant of the quotient $G$-space $G/H$. We show that for any infinite
$G$-space $c_{rt} \ge 1/2$. We say that $G/H$ is very small if $c_{rt}<1$. For a
normal subgroup $H$ the quotient space $G/H$ is very small if and only if it is finite.
However, we give examples of infinite very small $G$-spaces. We show also that critical
constants for recurrence can be used to estimate the growth of groups as well as the
drift for random walks on groups.
[1] A. Avez, “Entropie des groupes de type fini”, C. R. Acad. Sc. Paris, Sér. A 275 (1972), p. 1363-1366 MR 324741 | Zbl 0252.94013
[2] A. Avez, “Théorème de Choquet-Deny pour les groupes à croissance non exponentielle”, C. R. Acad. Sci. Paris, Sér. A 279 (1974), p. 25-28 MR 353405 | Zbl 0292.60100
[3] A. Avez, Croissance des groupes de type fini et fonctions harmoniques, Théorie ergodique, Lecture Notes in Math, Springer, Berlin, 1976, p. 35-49 Zbl 0368.60011
[4] A. Avez, Harmonic functions on groups, Differential geometry and relativity, Mathematical Phys. and Appl. Math., Reidel, Dordrecht, 1976, p. 27-32 Zbl 0345.31004
[5] L. Bartholdi, “The growth of Grigorchuk's torsion group”, Internat. Math. Res. Notices (1998) no. 20, p. 1049-1054 MR 1656258 | Zbl 0942.20027
[6] L. Bartholdi, “Lower bounds on the growth of a group acting on the binary rooted tree”, Internat. J. Algebra Comput. 11 (2001) no. 1, p. 73-88 Article | MR 1818662 | Zbl 1028.20025
[7] P. Baldi, N. Lohoué & J. Peyrière, “Sur la classification des groupes récurrents”, C. R. Acad. Sci. Paris, Sér. A-B 285 (1987) no. 16 MR 518008 | Zbl 0376.60072
[8] Y. Derriennic, Quelques applications du théorème ergodique sous-additif, Asterisque, 1980, p. 183-201 Zbl 0446.60059
[9] A. Dyubina, “An example of growth rate for random walk on group”, Russian Math. Surveys 54 (1999) no. 5, p. 159-160 MR 1741670 | Zbl 0964.60506
[10] A. Erschler (Dyubina), “On the asymptotics of drift”, Zapiski Sem. POMI 283 (2001), p. 251-257 Zbl 02182854
[11] A. Erschler (Dyubina), “Drift and entropy growth for random walk on groups”, Russian Math. Surveys 56 (2001) no. 3, p. 179-180 MR 1859741 | Zbl 1026.60004
[12] A. Erschler, “Boundary behavior for groups of subexponetial growth”, to appear in Ann. of Math. Zbl 1089.20025
[13] A. Erschler, “Drift and entropy growth for random walks on groups”, Annals of Probability 31 (2003) no. 3, p. 1193-1204 Article | MR 1988468 | Zbl 1043.60005
[14] W. Feller, An introduction to probability theory and its applications II, Wiley Series in Probability and Mathematical Statistics, John Wiley and Sons, New York, 1971 MR 270403 | Zbl 0219.60003
[15] R. I. Grigorchuk, “On Burnside's problem on periodic groups, Funct. Anal. Appl.”, 14 (1980), p. 41-43 Zbl 0595.20029
[16] R. I. Grigorchuk, “Degrees of growth of finitely generated groups, and the theory of invariant mean”, Math USSR Izv 25 (1985) no. 2, p. 259-300 Article | MR 764305 | Zbl 0583.20023
[17] R. I. Grigorchuk, “Groups with intermediate growth function and their applications”, Doctoral Thesis, 1985
[18] Y. Guivarch, Sur la loi des grands nombres et le rayon spectral d'une marche aléatoire, in Conference on Random Walks (Kleebach, 1979), Asterisque, Soc. Math. France, 1980, p. 47-98 Zbl 0448.60007
[19] Y. Guivarch, Marches aléatoires sur les groupes, Development of Mathematics, Birkhauser, p. 1950-2000 Zbl 0967.60057
[20] V. A. Kaimanovich & A. M. Vershik, “Random walks on discrete groups: boundary and entropy”, The Annals of Probability 11 (1983) no. 3, p. 457-490 Article | MR 704539 | Zbl 0641.60009
[21] Yu. G. Leonov & Yu. G., “On a lower bound for the growth of a 3-generator 2-group”, Mat. Sb. 192 (2001) no. 11, p. 77-92 MR 1886371 | Zbl 1031.20024
[22] A. Lubotzky, Cayley graphs: eigenvalues, expanders and random walks, Surveys in Combinatorics, Cambridge Univ. Press, 1995, p. 155-189 Zbl 0835.05033
[23] R. Muchnik & I. Pak, “On growth of Grigorchuk groups”, Internat. J. Algebra Comput. 11 (2001) no. 1, p. 1-17 Article | MR 1818659 | Zbl 1024.20031
[24] F. Spitzer, Principles of random walk, Van Nostrand, Princeton, 1964 MR 171290 | Zbl 0119.34304
[25] N. Th. Varoupoulos, “Théorie du potentiel sur des groupes et variétés”, C. R. Acad. Sci. Paris, Série I 302 (1986), p. 203-205 MR 832044 | Zbl 0605.31005
[26] N. Th. Varopoulos, L. Saloff-Coste & T. Coulhon, Analysis and geometry on groups, Cambridge Tracts in Mathematics 100, Cambridge University Press, Cambridge, 1992 MR 1218884 | Zbl 0813.22003
[27] W. Woess, Random walks on infinite graphs and groups, Cambr. Univ. Press, 2000 MR 1743100 | Zbl 0951.60002
Anna Erschler
Critical constants for recurrence of random walks on $G$-spaces
(Constantes critiques pour la récurrence des marches aléatoires sur les $G$-espace.)
Annales de l'institut Fourier, 55 no. 2 (2005), p. 493-509, doi: 10.5802/aif.2105
Article PDF | Reviews MR 2147898 | Zbl 02171516 | 1 citation in Cedram
Class. Math.: 20F65, 20E08, 60B15
Keywords: growth of groups, Grigorchuk groups, branch groups, random walks, recurrence, drift
Résumé - Abstract
Bibliography
[2] A. Avez, “Théorème de Choquet-Deny pour les groupes à croissance non exponentielle”, C. R. Acad. Sci. Paris, Sér. A 279 (1974), p. 25-28 MR 353405 | Zbl 0292.60100
[3] A. Avez, Croissance des groupes de type fini et fonctions harmoniques, Théorie ergodique, Lecture Notes in Math, Springer, Berlin, 1976, p. 35-49 Zbl 0368.60011
[4] A. Avez, Harmonic functions on groups, Differential geometry and relativity, Mathematical Phys. and Appl. Math., Reidel, Dordrecht, 1976, p. 27-32 Zbl 0345.31004
[5] L. Bartholdi, “The growth of Grigorchuk's torsion group”, Internat. Math. Res. Notices (1998) no. 20, p. 1049-1054 MR 1656258 | Zbl 0942.20027
[6] L. Bartholdi, “Lower bounds on the growth of a group acting on the binary rooted tree”, Internat. J. Algebra Comput. 11 (2001) no. 1, p. 73-88 Article | MR 1818662 | Zbl 1028.20025
[7] P. Baldi, N. Lohoué & J. Peyrière, “Sur la classification des groupes récurrents”, C. R. Acad. Sci. Paris, Sér. A-B 285 (1987) no. 16 MR 518008 | Zbl 0376.60072
[8] Y. Derriennic, Quelques applications du théorème ergodique sous-additif, Asterisque, 1980, p. 183-201 Zbl 0446.60059
[9] A. Dyubina, “An example of growth rate for random walk on group”, Russian Math. Surveys 54 (1999) no. 5, p. 159-160 MR 1741670 | Zbl 0964.60506
[10] A. Erschler (Dyubina), “On the asymptotics of drift”, Zapiski Sem. POMI 283 (2001), p. 251-257 Zbl 02182854
[11] A. Erschler (Dyubina), “Drift and entropy growth for random walk on groups”, Russian Math. Surveys 56 (2001) no. 3, p. 179-180 MR 1859741 | Zbl 1026.60004
[12] A. Erschler, “Boundary behavior for groups of subexponetial growth”, to appear in Ann. of Math. Zbl 1089.20025
[13] A. Erschler, “Drift and entropy growth for random walks on groups”, Annals of Probability 31 (2003) no. 3, p. 1193-1204 Article | MR 1988468 | Zbl 1043.60005
[14] W. Feller, An introduction to probability theory and its applications II, Wiley Series in Probability and Mathematical Statistics, John Wiley and Sons, New York, 1971 MR 270403 | Zbl 0219.60003
[15] R. I. Grigorchuk, “On Burnside's problem on periodic groups, Funct. Anal. Appl.”, 14 (1980), p. 41-43 Zbl 0595.20029
[16] R. I. Grigorchuk, “Degrees of growth of finitely generated groups, and the theory of invariant mean”, Math USSR Izv 25 (1985) no. 2, p. 259-300 Article | MR 764305 | Zbl 0583.20023
[17] R. I. Grigorchuk, “Groups with intermediate growth function and their applications”, Doctoral Thesis, 1985
[18] Y. Guivarch, Sur la loi des grands nombres et le rayon spectral d'une marche aléatoire, in Conference on Random Walks (Kleebach, 1979), Asterisque, Soc. Math. France, 1980, p. 47-98 Zbl 0448.60007
[19] Y. Guivarch, Marches aléatoires sur les groupes, Development of Mathematics, Birkhauser, p. 1950-2000 Zbl 0967.60057
[20] V. A. Kaimanovich & A. M. Vershik, “Random walks on discrete groups: boundary and entropy”, The Annals of Probability 11 (1983) no. 3, p. 457-490 Article | MR 704539 | Zbl 0641.60009
[21] Yu. G. Leonov & Yu. G., “On a lower bound for the growth of a 3-generator 2-group”, Mat. Sb. 192 (2001) no. 11, p. 77-92 MR 1886371 | Zbl 1031.20024
[22] A. Lubotzky, Cayley graphs: eigenvalues, expanders and random walks, Surveys in Combinatorics, Cambridge Univ. Press, 1995, p. 155-189 Zbl 0835.05033
[23] R. Muchnik & I. Pak, “On growth of Grigorchuk groups”, Internat. J. Algebra Comput. 11 (2001) no. 1, p. 1-17 Article | MR 1818659 | Zbl 1024.20031
[24] F. Spitzer, Principles of random walk, Van Nostrand, Princeton, 1964 MR 171290 | Zbl 0119.34304
[25] N. Th. Varoupoulos, “Théorie du potentiel sur des groupes et variétés”, C. R. Acad. Sci. Paris, Série I 302 (1986), p. 203-205 MR 832044 | Zbl 0605.31005
[26] N. Th. Varopoulos, L. Saloff-Coste & T. Coulhon, Analysis and geometry on groups, Cambridge Tracts in Mathematics 100, Cambridge University Press, Cambridge, 1992 MR 1218884 | Zbl 0813.22003
[27] W. Woess, Random walks on infinite graphs and groups, Cambr. Univ. Press, 2000 MR 1743100 | Zbl 0951.60002
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