Bounding the Complexity of Replica Symmetry Breaking for Spherical Spin Glasses

Abstract

In this paper, we study the Crisanti-Sommers variational problem, which is a variational formula for the free energy of spherical mixed $p$-spin glasses. We begin by computing the dual of this problem using a min-max argument. We find that the dual is a 1-D problem of obstacle type, where the obstacle is related to the covariance structure of the underlying process. This approach yields an alternative way to understand Replica Symmetry Breaking at the level of the variational problem through topological properties of the coincidence set of the optimal dual variable. Using this duality, we give an algorithm to reduce this a priori infinite dimensional variational problem to a finite dimensional one, thereby confining all possible forms of Replica Symmetry Breaking in these models to a finite parameter family. These results complement the authors' related results for the low temperature $\Gamma$-limit of this variational problem. We briefly discuss the analysis of the Replica Symmetric phase using this approach.

References in corpus (4)

• On the overlap in the multiple spherical SK models
• Amorphous-amorphous transition and the two-step replica symmetry breaking phase
• Low Temperature Asymptotics in Spherical Mean Field Spin Glasses
• Comment on ``Spherical 2 + p spin-glass model: An analytically solvable model with a glass-to-glass transition''

Cited by in corpus (4)

• Spectral gap estimates in mean field spin glasses
• Free energy in multi-species mixed $p$-spin spherical models
• Crisanti-Sommers formula and simultaneous symmetry breaking in multi-species spherical spin glasses
• On Marginal Stability in Low Temperature Spherical Spin Glasses