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In many spin glass models, due to the symmetry among sites, any limitingjoint distribution of spins under the annealed Gibbs measure admits theAldous-Hoover representation encoded by a function$\sigma:[0,1]^4\to\{-1,+1\}$, and one can think of this function as a genericfunctional order parameter of the model. In a class of diluted models, and inthe Sherrington-Kirkpatrick model, we introduce novel perturbations of theHamiltonian that yield certain invariance and self-consistency equations forthis generic functional order parameter and we use these invariance propertiesto obtain representations for the free energy in terms of $\sigma$. In thesetting of the Sherrington-Kirkpatrick model, the self-consistency equationsimply that the joint distribution of spins is determined by the jointdistributions of the overlaps, and we give an explicit formula for $\sigma$under the Parisi ultrametricity hypothesis. In addition, we discuss someconnections with the Ghirlanda-Guerra identities and stochastic stability anddescribe the expected Parisi ansatz in the diluted models in terms of $\sigma$.

Spin glass models from the point of view of spin distributions