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, Tome 55 (2005) no. 2, pp. 493-509.

On introduit la notion de constante critique c rt pour les marches aléatoires sur les G-espaces. Pour un sous-groupe H dans un groupe de type fini G, la constante critique de la récurrence est un invariant asymptotique du G-espace G/H. On montre que pour chaque G-espace infini, c rt 1/2. On dit que G/H est très petit si c rt <1. Pour un sous-groupe distingué H l’espace quotient G/H est très petit si et seulement si il est fini. Cependant, on donne des exemples de G-espaces très petits et infinis. On montre également que la constante critique pour la récurrence peut être utilisée pour estimer la croissance de groupes et la vitesse de fuite des marches aléatoires.

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 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.

DOI : 10.5802/aif.2105
Classification : 20F65, 20E08, 60B15
Keywords: growth of groups, Grigorchuk groups, branch groups, random walks, recurrence, drift
Mot clés : croissance des groupes, groupes de Grigorchuk, groupes branches, marches aléatoires, récurrence, vitesse de fuite.
Erschler, Anna 1

1 Université Lille 1, UFR de Mathématiques, 59655 Villeneuve d'Ascq Cedex (FRANCE)
@article{AIF_2005__55_2_493_0,
     author = {Erschler, Anna},
     title = {Critical constants for recurrence of random walks on $G$-spaces},
     journal = {Annales de l'Institut Fourier},
     pages = {493--509},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {55},
     number = {2},
     year = {2005},
     doi = {10.5802/aif.2105},
     mrnumber = {2147898},
     zbl = {02171516},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2105/}
}
TY  - JOUR
AU  - Erschler, Anna
TI  - Critical constants for recurrence of random walks on $G$-spaces
JO  - Annales de l'Institut Fourier
PY  - 2005
SP  - 493
EP  - 509
VL  - 55
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2105/
DO  - 10.5802/aif.2105
LA  - en
ID  - AIF_2005__55_2_493_0
ER  - 
%0 Journal Article
%A Erschler, Anna
%T Critical constants for recurrence of random walks on $G$-spaces
%J Annales de l'Institut Fourier
%D 2005
%P 493-509
%V 55
%N 2
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2105/
%R 10.5802/aif.2105
%G en
%F AIF_2005__55_2_493_0
Erschler, Anna. Critical constants for recurrence of random walks on $G$-spaces. Annales de l'Institut Fourier, Tome 55 (2005) no. 2, pp. 493-509. doi : 10.5802/aif.2105. http://www.numdam.org/articles/10.5802/aif.2105/

[1] A. Avez Entropie des groupes de type fini, C. R. Acad. Sc. Paris, Sér. A, Volume 275 (1972), pp. 1363-1366 | MR | Zbl

[2] A. Avez Théorème de Choquet-Deny pour les groupes à croissance non exponentielle, C. R. Acad. Sci. Paris, Sér. A, Volume 279 (1974), pp. 25-28 | MR | Zbl

[3] A. Avez Croissance des groupes de type fini et fonctions harmoniques, Théorie ergodique, Actes Journées Ergodiques, Rennes, 1973/1974 (Lecture Notes in Math), Volume 532 (1976), pp. 35-49 | Zbl

[4] A. Avez Harmonic functions on groups, Differential geometry and relativity (Mathematical Phys. and Appl. Math.), Volume 3 (1976), pp. 27-32 | Zbl

[5] L. Bartholdi The growth of Grigorchuk's torsion group, Internat. Math. Res. Notices (1998) no. 20, pp. 1049-1054 | MR | Zbl

[6] L. Bartholdi Lower bounds on the growth of a group acting on the binary rooted tree, Internat. J. Algebra Comput., Volume 11 (2001) no. 1, pp. 73-88 | DOI | MR | Zbl

[7] P. Baldi; N. Lohoué; J. Peyrière Sur la classification des groupes récurrents, C. R. Acad. Sci. Paris, Sér. A-B, Volume 285 (1987) no. 16 | MR | Zbl

[8] Y. Derriennic Quelques applications du théorème ergodique sous-additif (Asterisque), Volume 74 (1980), pp. 183-201 | Numdam | Zbl

[9] A. Dyubina An example of growth rate for random walk on group, Russian Math. Surveys, Volume 54 (1999) no. 5, pp. 159-160 | MR | Zbl

[10] A. Erschler (Dyubina) On the asymptotics of drift, Zapiski Sem. POMI, Volume 283 (2001), pp. 251-257 | Zbl

[11] A. Erschler (Dyubina) Drift and entropy growth for random walk on groups, Russian Math. Surveys, Volume 56 (2001) no. 3, pp. 179-180 | MR | Zbl

[12] A. Erschler Boundary behavior for groups of subexponetial growth (to appear in Ann. of Math.) | Zbl

[13] A. Erschler Drift and entropy growth for random walks on groups, Annals of Probability, Volume 31 (2003) no. 3, pp. 1193-1204 | DOI | MR | Zbl

[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 | Zbl

[15] R. I. Grigorchuk On Burnside's problem on periodic groups, Funct. Anal. Appl., Volume 14 (1980), pp. 41-43 | MR | Zbl

[16] R. I. Grigorchuk Degrees of growth of finitely generated groups, and the theory of invariant mean, Math USSR Izv, Volume 25 (1985) no. 2, pp. 259-300 | DOI | MR | Zbl

[17] R. I. Grigorchuk Groups with intermediate growth function and their applications (1985) (Doctoral Thesis)

[18] Y. Guivarch Sur la loi des grands nombres et le rayon spectral d'une marche aléatoire (Asterisque), Volume 74 (1980), pp. 47-98 | Numdam | MR | Zbl

[19] Y. Guivarch; J.P. Pier Marches aléatoires sur les groupes (Development of Mathematics), pp. 1950-2000 | MR | Zbl

[20] V. A. Kaimanovich; A. M. Vershik Random walks on discrete groups: boundary and entropy, The Annals of Probability, Volume 11 (1983) no. 3, pp. 457-490 | DOI | MR | Zbl

[21] Yu. G. Leonov; Yu. G. On a lower bound for the growth of a 3-generator 2-group, Mat. Sb., Volume 192 (2001) no. 11, pp. 77-92 | MR | Zbl

[22] A. Lubotzky; P .Rowlenson Cayley graphs: eigenvalues, expanders and random walks (Surveys in Combinatorics) (1995), pp. 155-189 | MR | Zbl

[23] R. Muchnik; I. Pak On growth of Grigorchuk groups, Internat. J. Algebra Comput., Volume 11 (2001) no. 1, pp. 1-17 | DOI | MR | Zbl

[24] F. Spitzer Principles of random walk, Van Nostrand, Princeton, 1964 | MR | Zbl

[25] N. Th. Varoupoulos Théorie du potentiel sur des groupes et variétés, C. R. Acad. Sci. Paris, Série I, Volume 302 (1986), pp. 203-205 | MR | Zbl

[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 | Zbl

[27] W. Woess Random walks on infinite graphs and groups, Cambr. Univ. Press, 2000 | MR | Zbl

Cité par Sources :