Random matrix theory of the proximity effect in disordered wires

We study analytically the local density of states in a disordered normal-metal wire (N) at ballistic distance to a superconductor (S). Our calculation is based on a scattering-matrix approach, which concerns for wave-function localization in the normal metal, and extends beyond the conventional semiclassical theory based on Usadel and Eilenberger equations. We also analyze how a finite transparency of the NS interface modifies the spectral proximity effect and demonstrate that our results agree in the dirty diffusive limit with those obtained from the Usadel equation.