Critical behavior of mean-field spin glasses on a dilute random graph

We provide a rigorous strategy to find the critical exponents of the overlaps for dilute spin glasses, in the absence of an external field. Such a strategy is based on the expansion of a suitably perturbed average of the overlaps, which is used in the formulation of the free energy as the difference between a cavity part and the derivative of the free energy itself, considered as a function of the connectivity of the model. We assume the validity of certain reasonable approximations, equivalent to assuming a second-order transition, e. g. that higher powers of overlap monomials are of smaller magnitude near the critical point, of which we do not provide a rigorous proof.