Spin glass models with ferromagnetically biased couplings on the Bethe lattice: analytic solutions and numerical simulations

TL;DR

This paper analytically characterizes the zero-temperature phase diagram of spin glass models with ferromagnetic bias on the Bethe lattice, identifying three phases and analyzing phase transitions using the cavity method, complemented by numerical simulations.

Contribution

It provides the first analytic solutions for the phase diagram of ferromagnetically biased spin glasses on the Bethe lattice using 1RSB cavity method.

Findings

Identified three distinct phases: RS ferromagnetic, mixed, and unmagnetized spin glass.

Derived analytic expressions for the critical points of phase transitions.

Compared analytical results with numerical algorithms, highlighting their limitations.

Abstract

We derive the zero-temperature phase diagram of spin glass models with a generic fraction of ferromagnetic interactions on the Bethe lattice. We use the cavity method at the level of one-step replica symmetry breaking (1RSB) and we find three phases: A replica-symmetric (RS) ferromagnetic one, a magnetized spin glass one (the so-called mixed phase), and an unmagnetized spin glass one. We are able to give analytic expressions for the critical point where the RS phase becomes unstable with respect to 1RSB solutions: we also clarify the mechanism inducing such a phase transition. Finally we compare our analytical results with the outcomes of a numerical algorithm especially designed for finding ground states in an efficient way, stressing weak points in the use of such numerical tools for discovering RSB effects. Some of the analytical results are given for generic connectivity.