Spin distributions for generic spherical spin glasses

This paper investigates spin distributions for a generic spherical p-spin model; we give a representation of spin distributions in terms of a stochastic process. In order to do this, we find a series of invariance principles analogous to the cavity equations used for the hypercube. The construction of these equations is a multistep process; we first change coordinates to write the sphere as a product space S-N-(R2) x [-C, C] for some cutoff parameters R and C. We then construct an auxiliary Hamiltonian that writes the effect of the final spin as an independent Gaussian process, e.g., H-N approximate to HN-1 + Y-N + Z(N) with Y-N and Z(N) decoupled from HN-1. The error rate in this approximation decays as O(C-4/R-2). The final step is to justify the double limit lim(C ->infinity )lim(R ->infinity) to derive exact results; this double limit can be interpreted as a novel renormalization procedure.