Sharp lower bounds for the asymptotic entropy of symmetric random walks

TL;DR

This paper establishes precise inequalities relating entropy, spectral radius, and drift for symmetric random walks on groups with finite second moments, revealing their deep interconnections and rigidity in equality cases.

Contribution

It provides sharp bounds linking entropy, spectral radius, and drift, improving previous inequalities and exploring their relation to group volume growth and rigidity in equality cases.

Findings

Sharp inequalities between entropy, spectral radius, and drift.

Relations between these quantities and group volume growth.

Rigidity results for cases of equality in the inequalities.

Abstract

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.