Homomorphisms to $ℝ$ constructed from random walks

Erschler, Anna; Karlsson, Anders

Annales de l'Institut Fourier, Tome 60 (2010) no. 6, pp. 2095-2113.

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.

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.

Classification : 20F69, 60B15, 60G50
Keywords: Random walks on groups, Liouville type theorems, growth of harmonic functions, homomorphisms to $\mathbb{R}$, groups of intermediate growth, entropy, drift, Gaussian estimates
Mot clés : marche aléatoires sur les groupes, théorèmes à la Liouville, croissance de fonctions harmoniques, homomorphismes dans $\mathbb{R}$, groupes de croissance intermédiaire, entropie, vitesse de fuite, estimations gaussiennes

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