# Random walks on groups: entropy, drift and volume growth.

vendredi 22 mai 2015, 9h30 - 10h30

Associated with random walks on groups, there are three fundamental
quantities: entropy, drift (escape speed) and volume growth
(exponential growth rate of the group).
The fundamental inequality due to Guivarc’h tells that the entropy
times the drift does not exceed the volume growth.
Vershik (2000) asked about the genuine equality case.
We focus on hyperbolic groups, and characterise the equality case;
namely, the equality holds if and only if the harmonic measure and a natural measure on the boundary are equivalent.
We also show a « confinement phenomenon » of the random walk in the
strict inequality case.
I start with a history of this problem and mention about recent
progresses as well.
All the notions will be explained during the talk.