We introduce the notion of a critical constant for recurrence of random walks on -spaces. For a subgroup of a finitely generated group the critical constant is an asymptotic invariant of the quotient -space . We show that for any infinite -space . We say that is very small if . For a normal subgroup the quotient space is very small if and only if it is finite. However, we give examples of infinite very small -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.
On introduit la notion de constante critique pour les marches aléatoires sur les -espaces. Pour un sous-groupe dans un groupe de type fini , la constante critique de la récurrence est un invariant asymptotique du -espace . On montre que pour chaque -espace infini, . On dit que est très petit si . Pour un sous-groupe distingué l’espace quotient est très petit si et seulement si il est fini. Cependant, on donne des exemples de -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.
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
@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}, zbl = {02171516}, mrnumber = {2147898}, language = {en}, url = {https://aif.centre-mersenne.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 - https://aif.centre-mersenne.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 https://aif.centre-mersenne.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, Volume 55 (2005) no. 2, pp. 493-509. doi : 10.5802/aif.2105. https://aif.centre-mersenne.org/articles/10.5802/aif.2105/
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