Groups, Geometry, and Dynamics


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Volume 9, Issue 3, 2015, pp. 711–735
DOI: 10.4171/GGD/325

Published online: 2015-10-29

Sharp lower bounds for the asymptotic entropy of symmetric random walks

Sébastien Gouëzel[1], Frédéric Mathéus[2] and François Maucourant[3]

(1) Université de Rennes I, France
(2) Université de Bretagne-Sud, Vannes, France
(3) Université de Rennes I, France

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.

Keywords: Random walk, countable group, entropy, spectral radius, drift, volume growth, Poisson boundary

Gouëzel Sébastien, Mathéus Frédéric, Maucourant François: Sharp lower bounds for the asymptotic entropy of symmetric random walks. Groups Geom. Dyn. 9 (2015), 711-735. doi: 10.4171/GGD/325