The Sherrington-Kirkpatrick model near T_c and near T=0

TL;DR

This paper investigates the Sherrington-Kirkpatrick model's behavior near the critical temperature and near zero temperature, deriving expansions and analyzing eigenvalue structures to understand phase transitions and solution regimes.

Contribution

It provides a detailed expansion of the replica free energy near T_c and characterizes the eigenvalue spectrum at low temperatures, revealing a droplet-like regime.

Findings

Replica free energy expansion near T_c up to 4th order

Eigenvalue bands maintain structure near T=0 for certain x ranges

Collapse of eigenvalue bands into two eigenvalues at very low T

Abstract

Some recent results concerning the Sherrington-Kirkpatrick model are reported. For $T$ near the critical temperature $T_c$, the replica free energy of the Sherrington-Kirkpatrick model is taken as the starting point of an expansion in powers of $\delta Q_{ab} = (Q_{ab} - Q_{ab}^{\rm RS})$ about the Replica Symmetric solution $Q_{ab}^{\rm RS}$. The expansion is kept up to 4-th order in $\delta{\bm Q}$ where a Parisi solution $Q_{ab} = Q(x)$ emerges, but only if one remains close enough to $T_c$. For $T$ near zero we show how to separate contributions from $x\ll T\ll 1$ where the Hessian maintains the standard structure of Parisi Replica Symmetry Breaking with bands of eigenvalues bounded below by zero modes. For $T\ll x \leq 1$ the bands collapse and only two eigenvalues, a null one and a positive one, are found. In this region the solution stands in what can be called a {\sl droplet-like} regime.