Irreducible free energy expansion and overlaps locking in mean field spin glasses

Following the works of [9],[11], we introduce a diagrammatic formulation for a cavity field expansion around the critical temperature. This approach allows us to obtain a theory for the overlap’s fluctuations and, in particular, the linear part of the Ghirlanda-Guerra relationships (GG) (often called Aizenman-Contucci polynomials (AC)) in a very simple way. We show moreover how these constraints are ”superimposed” by the symmetry of the model with respect to the restriction required by thermodynamic stability. Within this framework it is possible to expand the free energy in terms of these irreducible overlaps fluctuations and in a form that simply put in evidence how the complexity of the solution is related to the complexity of the entropy. key words: Cavity field, Ghirlanda-Guerra, stochastic stability, Aizenman-Contucci polynomials. King’s College London, Department of Mathematics, Strand, London WC2R 2LS, United Kingdom, and Dipartimento di Fisica, Università di Roma “La Sapienza” Piazzale Aldo Moro 2, 00185 Roma, Italy,