Phase transitions in the distribution of the Andreev conductance of   superconductor-metal junctions with multiple transverse modes
  We compute analytically the full distribution of Andreev conductance $G_{\mathrm{NS}}$ of a metal-superconductor interface with a large number $N_c$ of transverse modes, using a random matrix approach. The probability distribution $\mathcal{P}(G_{\mathrm{NS}},N_c)$ in the limit of large $N_c$ displays a Gaussian behavior near the average value $<G_{\mathrm{NS}}>= (2-\sqrt{2}) N_c$ and asymmetric power-law tails in the two limits of very small and very large $G_{\mathrm{NS}}$. In addition, we find a novel third regime sandwiched between the central Gaussian peak and the power law tail for large $G_{\mathrm{NS}}$. Weakly non-analytic points separate these four regimes---these are shown to be consequences of three phase transitions in an associated Coulomb gas problem. 
Kedar Damle, Satya N. Majumdar, Vikram Tripathi, Pierpaolo Vivo
cond-mat.mes-hall cond-mat.stat-mech math-ph math.MP