Landscape statistics of the p-spin Ising model

TL;DR

This paper analytically investigates the statistical properties of local optima in the p-spin Ising model, revealing how overlaps and the number of optima depend on parameters like p and magnetic field.

Contribution

It provides new analytical calculations of the number and overlap of local optima in the p-spin Ising model under external magnetic fields.

Findings

Overlap q_t is discontinuous for p>2 and small h.

The size of the overlap jump increases with p.

The overlap jump decreases with increasing magnetic field h.

Abstract

The statistical properties of the local optima (metastable states) of the infinite range Ising spin glass with p-spin interactions in the presence of an external magnetic field h are investigated analytically. The average number of optima as well as the typical overlap between pairs of identical optima are calculated for general p. Similarly to the thermodynamic order parameter, for p>2 and small h the typical overlap q_t is a discontinuous function of the energy. The size of the jump in q_t increases with p and decreases with h, vanishing at finite values of the magnetic field.