2006/09/13 by Dan Y. C. Shiber, Dan Shiber, Shiber, Dan
Computer Science · Mathematics · Physics and Astronomy · #15A52 Random matrices #FOS: Mathematics #Morphological variations and asymmetry #Operator Algebras (math.OA) #Probability (math.PR) #Statistical Mechanics and Entropy #Topological and Geometric Data Analysis #math.OA #math.PR #msc:15A52
paper · pdf · doi:10.48550/arxiv.math/0609372
29 pages, no figures; corrected typos, a few sections revised
openalex publication_date 2006/09/13 · arxiv created 2007/04/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.