  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.
