  The core idea of stochastic stability is that thermodynamic observables must
be robust under small (random) perturbations of the quenched Gibbs measure.
Combining this idea with the cavity field technique, which aims to measure the
free energy increment under addition of a spin to the system, we sketch how to
write a stochastic stability approach to diluted mean field spin glasses which
explicitly gives overlap constraints as the outcome. We then show that, under
minimal mathematical assumptions and for gauge invariant systems (namely those
with even Ising interactions), it is possible to "reverse" the idea of
stochastic stability and use it to derive a broad class of constraints on the
unperturbed quenched Gibbs measure. This paper extends a previous study where
we showed how to derive (linear) polynomial identities from the "energy"
contribution to the free energy, while here we focus on the consequences of
"entropic" constraints. Interestingly, in diluted spin glasses, the entropic
approach generates more identities than those found by the energy route or
other techniques. The two sets of identities become identical on a fully
connected topology, where they reduce to the ones derived by Aizenman and
Contucci.
