  Let X be a compact complex manifold equipped with a smooth (but not
necessarily positive) closed form theta of one-one type. By a well-known
envelope construction this data determines a canonical theta-psh function u
which is not two times differentiable, in general. We introduce a family of
regularizations of u, parametrized by a positive number beta, defined as the
smooth solutions of complex Monge-Ampere equations of Aubin-Yau type. It is
shown that, as beta tends to infinity, the regularizations converge to the
envelope u in the strongest possible Holder sense. A generalization of this
result to the case of a nef and big cohomology class is also obtained. As a
consequence new PDE proofs are obtained for the regularity results for
envelopes in [14] (which, however, are weaker than the results in [14] in the
case of a non-nef big class). Applications to the regularization problem for
quasi-psh functions and geodesic rays in the closure of the space of Kahler
metrics are given. As briefly explained there is a statistical mechanical
motivation for this regularization procedure, where beta appears as the inverse
temperature. This point of view also leads to an interpretation of the
regularizations as transcendental Bergman metrics.
