  We investigate the analogy between the large N expansion in normal matrix
models and the asymptotic expansion of the determinant of the Hilb map,
appearing in the study of critical metrics on complex manifolds via projective
embeddings. This analogy helps to understand the geometric meaning of the
expansion of matrix model free energy and its relation to gravitational
effective actions in two dimensions. We compute the leading terms of the free
energy expansion in the pure bulk case, and make some observations on the
structure of the expansion to all orders. As an application of these results,
we propose an asymptotic formula for the Liouville action, restricted to the
space of the Bergman metrics.
