The replica limit of unitary matrix integrals

We investigate the replica trick for the microscopic spectral density, rho (S) (x), of the Euclidean QCD Dirac operator. Our starting point is the low -energy limit of the QCD partition function for n fermionic flavors (or rep licas) in the sector of topological charge v. In the domain of the smallest eigenvalues, this partition function is simply given by a U(n) unitary mat rix integral. We show, that the asymptotic expansion of rho (S) (x) for x - -> infinity is obtained from the n --> 0 limit of this integral. The smooth contributions to this series are obtained from an expansion about the repl ica symmetric saddle-point, whereas the oscillatory terms follow from an ex pansion about a saddle-point that breaks the replica symmetry. For v = 0 we recover the small-x logarithmic singularity of the resolvent by means of t he replica trick. For half integer v, when the saddle point expansion of th e U(n) integral terminates, the replica trick reproduces the exact analytic al result. In all other cases only an asymptotic series that does not uniqu ely determine the microscopic spectral density is obtained. We argue that b osonic replicas fail to reproduce the microscopic spectral density. In all cases, the exact answer is obtained naturally by means of the supersymmetri c method. (C) 2001 Elsevier Science B.V. All rights reserved.