Quenched Randomness in the Two-Dimensional Ferromagnetic Planar Model

A model of a disordered planar model is introduced and solved with use of the replica method. The model is designed to mimic the inhomogeneities known to exist in helium and granular superconducting films. The results indicate that at onset, the universal results, predicted on the basis of renormalization-group analysis, hold to lowest order in the parameter measuring the disorder. The disorder has the important effect of increasing the vortex pairs couplings, resulting in an initial increase of the critical temperature. Possible consequences of the results to related experiments are also presented.