Free transport-entropy inequalities for non-convex potentials and application to concentration for random matrices

Talagrand's inequalities make a link between two fundamentals concepts of probability: transport of measures and entropy. The study of the counterpart of these inequalities in the context of free probability has been initiated by Biane and Voiculescu and later extended by Hiai, Petz and Ueda for convex potentials. In this work, we prove a free analogue of a result of Bobkov and G\"otze in the classical setting, thus providing free transport-entropy inequalities for a very natural class of measures appearing in random matrix theory. These inequalities are weaker than the ones of Hiai, Petz and Ueda but still hold beyond the convex case. We then use this result to get a concentration estimate for $\beta$-ensembles under mild assumptions on the potential.