  The heat kernel on the symmetric space of positive definite Hermitian
matrices is used to endow the spaces of Bergman metrics of degree k on a
Riemann surface M with a family of probability measures depending on a choice
of the background metric. Under a certain matrix-metric correspondence, each
positive definite Hermitian matrix corresponds to a Kahler metric on M. The one
and two point functions of the random metric are calculated in a variety of
limits as k and t tend to infinity. In the limit when the time t goes to
infinity the fluctuations of the random metric around the background metric are
the same as the fluctuations of random zeros of holomorphic sections. This is
due to the fact that the random zeros form the boundary of the space of Bergman
metrics.
