On mean field equations for spin glasses

Abstract

In this paper we study rigorously the random Ising model on a Cayley tree in the limit of infinite coordination number z → 8. An iterative scheme is developed relating mean magnetizations and mean square magnetizations of successive shells far removed from the surface of the lattice. In this way we obtain local properties of the model in the (thermodynamic) limit of an infinite number of shells. When the coupling constants are independent Gaussian random variables the SK expressions emerge as stable fixed points of our scheme and provide a valid local mean-field theory of spin glasses in which negative local entropy (at low temperatures) while perfectly possible mathematically may still perh..