  This is mainly a survey, explaining how the probabilistic (statistical
mechanical) construction of Kahler-Einstein metrics on compact complex
manifolds, introduced in a series of works by the author, naturally arises from
classical approximation and interpolation problems in complex n-space. A fair
amount of background material is included. Along the way, the results are
generalized to the non-compact setting of complex n-space. This yields a
probabilistic construction of Kahler solutions to Einstein's equations in
complex n-space, with cosmological constant -beta, from a gas of interpolation
nodes in equilibrium at positive inverse temperature -beta. In the infinite
temperature limit, solutions to the Calabi-Yau equation are obtained. In the
opposite zero-temperature the results may be interpreted as "transcendental"
analogs of classical asymptotics for orthogonal polynomials, with the inverse
temperature playing the role of the degree of a polynomial.
