The mobility edge as a spin-glass problem

An analogy is considered between long-range properties of the Green function G of an electron moving in a random potential, near the mobility edge, and those of spin-spin correlation functions, obtained for a random Ginzburg-Landau Gaussian model. The absence of a 'ferromagnetic' long-range order in the latter model is related to the short range of the average G. The average squared modulus may become long ranged. This long range is analogous to a 'spin-glass' like phase. This 'spin-glass' transition deviates from mean-field theory for dimensionalities d<4. Renormalisation group, the epsilon expansion and the n to 0 replica trick are used to analyse the appropriate fixed points. For few impurities, no fixed point can be reached, probably because no localisation edge exists. For larger disorder, the 'isotropic', n=0 fixed point may be reached, and is interpreted as probably leading to percolation. For still larger disorder, the Anderson transition may result.