Hierarchical solutions of the Sherrington-Kirkpatrick model: Exact asymptotic behavior near the critical temperature

TL;DR

This paper investigates the asymptotic behavior of the replica-symmetry-breaking solutions in the Sherrington-Kirkpatrick spin glass model near the critical temperature, revealing stability properties and the transition from discrete to continuous hierarchies.

Contribution

It provides a general scheme for deriving exact asymptotics of replica-symmetry-breaking solutions with any number of hierarchies in the SK model.

Findings

Finite hierarchies are unstable

Infinite hierarchies are marginally stable

Discrete solutions converge to continuous scheme as hierarchies increase

Abstract

We analyze the replica-symmetry-breaking construction in the Sherrington-Kirkpatrick model of a spin glass. We present a general scheme for deriving an exact asymptotic behavior near the critical temperature of the solution with an arbitrary number of discrete hierarchies of the broken replica symmetry. We show that all solutions with finite-many hierarchies are unstable and only the scheme with infinite-many hierarchies becomes marginally stable. We show how the solutions from the discrete replica-symmetry-breaking scheme go over to the continuous one with increasing the number of hierarchies.