  Exact microscopic spectral correlation functions are derived by means of the
replica limit of the Toda lattice equation. We consider both Hermitian and
non-Hermitian theories in the Wigner-Dyson universality class (class A) and in
the chiral universality class (class AIII). In the Hermitian case we rederive
two-point correlation functions for class A and class AIII as well as several
one-point correlation functions in class AIII. In the non-Hermitian case the
spectral density of non-Hermitian complex random matrices in the weak
non-Hermiticity limit is obtained directly from the replica limit of the Toda
lattice equation. In the case of class A, this result describes the spectral
density of a disordered system in a constant imaginary vector potential (the
Hatano-Nelson model) which is known from earlier work. New results are obtained
for the spectral density in the weak non-Hermiticity limit of a quenched chiral
random matrix model at nonzero chemical potential. These results apply to the
ergodic or $\epsilon$ domain of quenched QCD at nonzero chemical potential. The
spectral density obtained is different from the result derived by Akemann for a
closely related model, which is given by the leading order asymptotic expansion
of our result. In all cases, the replica limit of the Toda lattice equation
explains the factorization of spectral one- and two-point functions into a
product of a bosonic (noncompact integral) and a fermionic (compact integral)
partition function. We conclude that the fermionic, the bosonic and the
supersymmetric partition functions are all part of a single integrable
hierarchy. This is the reason that it is possible to obtain the supersymmetric
partition function, and its derivatives, from the replica limit of the Toda
lattice equation.
