  Exact solvability is claimed for nonlinear replica sigma models derived in
the context of random matrix theories. Contrary to other approaches reported in
the literature, the framework outlined does not rely on traditional "replica
symmetry breaking" but rests on a previously unnoticed exact relation between
replica partition functions and Painleve transcendents. While expected to be
applicable to matrix models of arbitrary symmetries, the method is used to
treat fermionic replicas for the Gaussian unitary ensemble (GUE), chiral GUE
(symmetry classes A and AIII in Cartan classification) and Ginibre's ensemble
of complex non-Hermitean random matrices. Further applications are briefly
discussed.
