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This work is devoted to direct mass transportation proofs of families of functional inequalities in the context of one-dimensional free probability, avoiding random matrix approximation. The inequalities include the free form of the transportation, Log-Sobolev, HWI interpolation and Brunn-Minkowski inequalities for strictly convex potentials. Sharp constants and some extended versions are put forward. The paper also addresses two versions of free Poincaré inequalities and their interpretation in terms of spectral properties of Jacobi operators. The last part establishes the corresponding inequalities for measures on R + with the reference example of the Marcenko-Pastur distribution.

In this paper we discuss general tridiagonal matrix models which are natural extensions of the ones given in [3] and [4]. We prove here the convergence of the distribution of the eigenvalues and compute the limiting distributions in some particular cases. We also discuss the limit of fluctuations, which, in a general context, turn out to be Gaussian. For the case of several random matrices, we prove the convergence of the joint moments and the convergence of the fluctuations to a Gaussian family.The methods involved are based on an elementary result on sequences of real numbers and a judicious counting of levels of paths.

Zipf’s law, Rank-frequency distribution, Synthetic language,

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