Information Geometry of Random Matrix Models

Details Information Geometry of Random Matrix Models Information Geometry of Random Matrix Models

2006-09-13

arxiv.org/abs/math/0609372v3

Mathematics

Operator Algebras

29 pages, no figures; corrected typos, a few sections revised

Scientific paper

In this paper we develop the theory of information geometry for single random matrix models, with two goals: proving a Cramer-Rao theorem for estimators on random matrices, and calculating the Legendre transform of pressure and entropy with respect to a metric duality. Consequently, in the large n limit we recover several quantities from free probability: Voiculescu's conjugate variable is the tangent vector to the GUE perturbation model, giving rise to a metric which turns out to be the free Fisher information measure; Hiai's Legendre transform of free pressure agrees with our Legendre transform of pressure; and Speicher's covariance of fluctuations naturally arises as the metric on the random matrix model obtained from the fluctuation functions.

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