Replica-symmetry breaking: discrete and continuous schemes in the Sherrington-Kirkpatrick model

TL;DR

This paper explores the hierarchical solutions of the Sherrington-Kirkpatrick model, showing how discrete schemes transition into the continuous Parisi scheme and analyzing solutions near the instability line.

Contribution

It derives stationarity equations for arbitrary hierarchies and elucidates the emergence of the continuous replica-symmetry breaking scheme.

Findings

Derivation of equations for arbitrary hierarchy levels.

Demonstration of the continuous scheme emergence from discrete hierarchies.

Asymptotic analysis near the de Almeida-Thouless line.

Abstract

We study hierarchies of replica-symmetry-breaking solutions of the Sherrington-Kirkpatrick model. Stationarity equations for order parameters of solutions with an arbitrary number of hierarchies are set and the limit to infinite number of hierarchical levels is discussed. In particular, we demonstrate how the continuous replica-symmetry breaking scheme of Parisi emerges and how the limit to infinite-many hierarchies leads to equations for the order-parameter function of the continuous solution. The general analysis is accompanied by an explicit asymptotic solution near the de Almeida-Thouless instability line in the nonzero magnetic field.