Percolation of Superconductivity

In case of superconductors whose electrons attract each other only if they are near certain centers, the question arises 'How many such centers are needed to make the ground state superconducting?' We shall examine it in the context of a random U Hubbard model. In short we study the case where U_i is -|U| and 0 with probability c and 1-c respectively on a lattice whose sites are labelled i using the Gorkov decoupling and the Coherent Potential Approximation (CPA). We argue that for this model there is a critical concentration c_0 below which the system is not a superconductor.