A SHARP SYMMETRIZED FREE TRANSPORT-ENTROPY INEQUALITY FOR THE SEMICIRCULAR LAW

In this paper, following the recent work of Fathi (2018) in the classical case, we provide by two different methods a sharp symmetrized free Talagrand inequality for the semicircular law, which improves the free TCI of Biane and Voiculescu (2000). The first proof holds only in the onedimensional case and has the advantage of providing a connection with the machinery of free moment maps introduced by Bahr and Boschert (2023) and a free reverse Log-Sobolev inequality. This case also and sheds light on a dual formulation via the free version of the functional Blaschke-Santaló inequality. The second proof gives the result in a multidimensional setting and relies on a random matrix approximation approach developed by Biane (2003), Hiai, Petz and Ueda (2004) combined with Fathi's inequality on Euclidean spaces.