Homomorphisms to constructed from random walks
Annales de l'Institut Fourier, Volume 60 (2010) no. 6, p. 2095-2113

We give a construction of homomorphisms from a group into the reals using random walks on the group. The construction is an alternative to an earlier construction that works in more general situations. Applications include an estimate on the drift of random walks on groups of subexponential growth admitting no nontrivial homomorphism to the integers and inequalities between the asymptotic drift and the asymptotic entropy. Some of the entropy estimates obtained have applications independent of the homomorphism construction, for example a Liouville-type theorem for slowly growing harmonic functions on groups of subexponential growth and on some groups of exponential growth.

Nous donnons une construction d’homomorphismes d’un groupe dans les nombres réels en utilisant une marche aléatoire sur le groupe. Cette construction est une alternative à une construction antécédente qui de plus s’applique dans des cas plus généraux. Les applications comprennent une estimation de la vitesse de fuite de marches aléatoires sur des groupes de croissance sous-exponentielle n’admettant pas d’homomorphismes non triviaux dans les nombres entiers et des inégalités entre la vitesse de fuite asymptotique et l’entropie asymptotique. Certaines des estimations d’entropie obtenues ont des applications indépendantes de la construction de l’homomorphisme, comme par exemple un théorème à la Liouville pour les fonctions harmoniques croissant lentement sur les groupes de croissance sous-exponentielle et certains groupes de croissance exponentielle.

DOI : https://doi.org/10.5802/aif.2577
Classification:  20F69,  60B15,  60G50
Keywords: Random walks on groups, Liouville type theorems, growth of harmonic functions, homomorphisms to , groups of intermediate growth, entropy, drift, Gaussian estimates
@article{AIF_2010__60_6_2095_0,
     author = {Erschler, Anna and Karlsson, Anders},
     title = {Homomorphisms to $\mathbb{R}$ constructed from random walks},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {60},
     number = {6},
     year = {2010},
     pages = {2095-2113},
     doi = {10.5802/aif.2577},
     mrnumber = {2791651},
     zbl = {1274.60015},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2010__60_6_2095_0}
}
Erschler, Anna; Karlsson, Anders. Homomorphisms to $\mathbb{R}$ constructed from random walks. Annales de l'Institut Fourier, Volume 60 (2010) no. 6, pp. 2095-2113. doi : 10.5802/aif.2577. http://www.numdam.org/item/AIF_2010__60_6_2095_0/

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