Critical constants for recurrence of random walks on G -spaces

Anna Erschler[1]

  • [1] Université Lille 1, UFR de Mathématiques, 59655 Villeneuve d'Ascq Cedex (FRANCE)

Annales de l’institut Fourier (2005)

  • Volume: 55, Issue: 2, page 493-509
  • ISSN: 0373-0956

Abstract

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We introduce the notion of a critical constant c r t 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 r t ≥ 1 / 2 . We say that G / H is very small if c r t < 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.

How to cite

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Erschler, Anna. "Critical constants for recurrence of random walks on $G$-spaces." Annales de l’institut Fourier 55.2 (2005): 493-509. <http://eudml.org/doc/116198>.

@article{Erschler2005,
abstract = {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 $c_\{rt\}&lt;1$. 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.},
affiliation = {Université Lille 1, UFR de Mathématiques, 59655 Villeneuve d'Ascq Cedex (FRANCE)},
author = {Erschler, Anna},
journal = {Annales de l’institut Fourier},
keywords = {growth of groups; Grigorchuk groups; branch groups; random walks; recurrence; drift; random walks on discrete groups; asymptotic invariants; asymptotic behavior; drift growth; entropy growth},
language = {eng},
number = {2},
pages = {493-509},
publisher = {Association des Annales de l'Institut Fourier},
title = {Critical constants for recurrence of random walks on $G$-spaces},
url = {http://eudml.org/doc/116198},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Erschler, Anna
TI - Critical constants for recurrence of random walks on $G$-spaces
JO - Annales de l’institut Fourier
PY - 2005
PB - Association des Annales de l'Institut Fourier
VL - 55
IS - 2
SP - 493
EP - 509
AB - 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 $c_{rt}&lt;1$. 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.
LA - eng
KW - growth of groups; Grigorchuk groups; branch groups; random walks; recurrence; drift; random walks on discrete groups; asymptotic invariants; asymptotic behavior; drift growth; entropy growth
UR - http://eudml.org/doc/116198
ER -

References

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