We investigate the asymptotic behaviour of the return probability of the random walk in finitely generated metabelian groups. For such groups with exponential volume growth, we obtain a characterization of metabelian groups whose return probability is the largest in purely algebraic terms, namely the Krull dimension of the group. Along the way, we give lower bounds on the return probability for metabelian groups with torsion derived subgroup, according to the dimension. We also establish a variation of the famous embedding theorem of Kaloujinine and Krasner for metabelian groups that respects the Krull dimension. Finally, we study specific sections of these groups, and use them to give upper bounds on the return probability in terms of the Krull dimension.
Dans cet article, on s’intéresse au comportement asymptotique de la probabilité de retour de la marche aléatoire dans un groupe métabélien de type fini. Pour de tels groupes à croissance exponentielle, on obtient une caractérisation de ceux dont la probabilité de retour est la plus grande en des termes purement algébriques, à l’aide de la dimension de Krull du groupe. Nous obtenons d’abord des bornes inférieures dépendant de la dimension de Krull sur la probabilité de retour des groupes métabéliens dont le sous-groupe dérivé est de torsion. On prouve également une variante respectant la dimension de Krull d’un théorème de Kaloujinine et Krasner pour les groupes métabéliens. Enfin, on étudie des sections particulières de ces groupes afin de donner des bornes supérieures sur la probabilité de retour en fonction de la dimension de Krull.
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Keywords: Krull dimension, metabelian groups, return probability
Mot clés : dimension de Krull, groupes métabéliens, probabilité de retour
Jacoboni, Lison 1
@article{AIF_2019__69_5_2121_0, author = {Jacoboni, Lison}, title = {Metabelian groups with large return probability}, journal = {Annales de l'Institut Fourier}, pages = {2121--2167}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {69}, number = {5}, year = {2019}, doi = {10.5802/aif.3291}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3291/} }
TY - JOUR AU - Jacoboni, Lison TI - Metabelian groups with large return probability JO - Annales de l'Institut Fourier PY - 2019 SP - 2121 EP - 2167 VL - 69 IS - 5 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3291/ DO - 10.5802/aif.3291 LA - en ID - AIF_2019__69_5_2121_0 ER -
%0 Journal Article %A Jacoboni, Lison %T Metabelian groups with large return probability %J Annales de l'Institut Fourier %D 2019 %P 2121-2167 %V 69 %N 5 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3291/ %R 10.5802/aif.3291 %G en %F AIF_2019__69_5_2121_0
Jacoboni, Lison. Metabelian groups with large return probability. Annales de l'Institut Fourier, Volume 69 (2019) no. 5, pp. 2121-2167. doi : 10.5802/aif.3291. https://aif.centre-mersenne.org/articles/10.5802/aif.3291/
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