Spin distributions for generic spherical spin glasses

Abstract

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 $S_{N-R^2}\times [-C,C]$ 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., $H_N\approx H_{N-1}+Y_N+Z_N$ with $Y_N$ and $Z_N$ decoupled from $H_{N-1}$. The error rate in this approximation decays as $O(\frac{C^4}{R^2})$. The final step is to justify the double limit ${lim}_{C\to \mathrm{\infty }}{lim}_{R\to \mathrm{\infty }}$ to derive exact results; this double limit can be interpreted as a novel renormalization procedure.