Critique of the replica trick

Abstract

It is shown that the replica trick fails to give the correct non-perturbative result for the two-point function S₂ of the Gaussian unitary ensemble of N*N random matrices. The failure arises from an incorrect description of the symmetries of the random-matrix system in the limit N to infinity . The correct description, which involves integration over both non-compact and compact degrees of freedom, is obtained by using the method of superfields. Some implications for the localisation transition in disordered electronic systems and the theory of the quantised Hall effect are suggested.