A Simple Approach to the Complexity of Mean Field Ising Spin Glasses

Abstract

The geometry of spherical spin glasses is relatively well understood, thanks in large part to the Kac-Rice formula. However, this formula does not directly apply to discrete spin systems, and much less is known mathematically about the geometry of Ising spin glasses. In this paper, we present a simple method for counting local optima and gapped states in a broad class of discrete spin glasses. To illustrate its scope, we compute the annealed complexities ("configurational entropies," in the physics language) of mixed Ising spin glasses with potentials, such as external fields, and of the Ghatak-Sherrington model. These computations recover several existing results in the mathematics and physics literature as special cases. We can also compute energy-constrained complexity, finding a critical energy threshold analogous to the spherical case, but for gapped states of pure Ising spin glasses. Finally, we use our method to establish a form of disorder persistence for gapped states, identifying a transition between $3$-spin and $4$-spin glasses.