A sharp symmetrized free transport-entropy inequality for the semicircular law

TL;DR

This paper establishes a sharp symmetrized free transport-entropy inequality for the semicircular law, improving previous results and connecting it with free moment maps, reverse Log-Sobolev inequalities, and matrix approximation methods.

Contribution

It provides two novel proofs of the inequality: one in one dimension linking to free moment maps and dual inequalities, and another in multiple dimensions using matrix approximation techniques.

Findings

Sharp symmetrized free Talagrand inequality for semicircular law

Connection with free moment maps and reverse Log-Sobolev inequality

Multidimensional extension via matrix approximation methods

Abstract

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 one-dimensional 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-Santalo 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.