The rate of escape for random walks on polycyclic and metabelian groups

Thompson, Russ

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 1, pp. 270-287.

Nous utilisons la notion de distorsion des sous-groupes afin de déterminer la vitesse de fuite (sous linéaire) d'une marche aléatoire simple sur une classe de groupes polycycliques, et nous montrons que cette vitesse est invariante par changement de générateurs pour ces groupes. Pour les groupes métabéliens, nous définissons une forme plus forte de distorsion des sous-groupes qui s'applique à des sous-groupes non finiment engendrés. Sous cette hypothèse, nous calculons la vitesse de fuite pour certaines marches aléatoires sur certains groupes abélien par cyclique via l'intermédiaire d'une comparaison avec la chute d'un tas de sable abélien dissipatif.

We use subgroup distortion to determine the rate of escape of a simple random walk on a class of polycyclic groups, and we show that the rate of escape is invariant under changes of generating set for these groups. For metabelian groups, we define a stronger form of subgroup distortion which applies to non-finitely generated subgroups. Under this hypothesis, we compute the rate of escape for certain random walks on some abelian-by-cyclic groups via a comparison to the toppling of a dissipative abelian sandpile.

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