Sharp lower bounds for the asymptotic entropy of symmetric random walks

Sébastien Gouëzel (1) , Frédéric Mathéus (2) , François Maucourant (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 LMBA - Laboratoire de Mathématiques de Bretagne Atlantique

The entropy, the spectral radius and the drift are important numerical quantities associated to random walks on countable groups. We prove sharp inequalities relating those quantities for walks with a finite second moment, improving upon previous results of Avez, Varopoulos, Carne, Ledrappier. We also deduce inequalities between these quantities and the volume growth of the group. Finally, we show that the equality case in our inequality is rather rigid.

Mots clés: Random walk; countable group; entropy; spectral radius; drift; volume growth; Poisson boundary

Domaines: Probabilités [math.PR]

Sébastien Gouëzel, Frédéric Mathéus, François Maucourant. Sharp lower bounds for the asymptotic entropy of symmetric random walks. Groups, Geometry, and Dynamics, 2015, 9 (3), pp.711-735. ⟨10.4171/GGD/325⟩. ⟨hal-00732487v3⟩