Free transportation cost inequalities via random matrix approximation

Abstract

Abstract. By means of random matrix approximation procedure, we reprove Biane and Voiculescu's free analog of Talagrand's transportation cost inequality for measures on R in a more general setup. Furthermore, we prove the free transportation cost inequality for measures on T as well by extending the method to special unitary random matrices.

Transport inequalities and Concentration of measure

“…The proof above is close but different from the proof by Hiai, Petz and Ueda [67] also involving large deviations. See also [78] for a direct transport proof of Biane-Voiculescu transport inequality, and [79] for a transport inequality involving the W 1 distance.…”

A (one-dimensional) free Brunn–Minkowski inequality

“…The pertinence of Hamilton-Jacobi equations in this investigation has been particularly emphasized in [5,11]. The aim of this Note is to proceed to a similar scheme in the context of one-dimensional free probability theory, using random matrix approximation following the recent investigations by Biane [2] and Hiai, Petz and Ueda [7,8]. We rely specifically on the large deviation asymptotics of spectral measures of unitary invariant Hermitian random matrices put forward by Voiculescu [16] and Ben Arous and Guionnet [1] (cf.…”

Free Transportation Cost Inequalities for Noncommutative Multi-Variables

“…In fact, in [10,11,12] we adopted the definition with a 1 2 -multiple constant so that the bounds of TCI's there and in the present paper are 2 times different.…”

“…Biane and D. Voiculescu [4] proved the free analogue of Talagrand's TCI for compactly supported measures on R, where the relative entropy is replaced by its free analogue and the Gaussian measure by the semicircular one. In [11,12] we developed the random matrix approximation method to obtain a slight generalization of Biane and Voiculescu's free TCI as well as its counterpart on the circle T. The free analogues of the LSI's on R and on T were also obtained in [3] and [11,13] by the same method.…”

“…Recently, M. Ledoux [17] used a similar random matrix technique to prove the free analogue of the Brunn-Minkowski inequality for measures on R, from which (together with the Hamilton-Jacobi approach) he gave short proofs of the free TCI and LSI for measures on R. Furthermore, his approach was shown in [10] to be still applicable for getting the free TCI in [12] for measures on T.…”