The one dimensional free Poincaré inequality

Ledoux, Michel; Popescu, Ionel

doi:10.1090/S0002-9947-2013-05830-8

The one dimensional free Poincaré inequality
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by Michel Ledoux and Ionel Popescu PDF

Trans. Amer. Math. Soc. 365 (2013), 4811-4849 Request permission

Abstract:

In this paper we discuss the natural candidate for the one dimensional free Poincaré inequality. Two main strong points sustain this candidacy. One is the random matrix heuristic and the other the relations with the other free functional inequalities, namely, the free transportation and Log-Sobolev inequalities. As in the classical case the Poincaré is implied by the others. This investigation is driven by a nice lemma of Haagerup which relates logarithmic potentials and Chebyshev polynomials. The Poincaré inequality revolves around the counting number operator for the Chebyshev polynomials of the first kind with respect to the arcsine law on $[-2,2]$. This counting number operator appears naturally in a representation of the minimum of the logarithmic energy with external fields discovered in Analyticity of the planar limit of a matrix model by S. Garoufalidis and the second author as well as in the perturbation of logarithmic energy with external fields, which is the essential connection between all these inequalities.

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