Low temperature solution of the Sherrington-Kirkpatrick model

TL;DR

This paper introduces a simple scaling ansatz for the full replica symmetry breaking solution of the SK model at low energies, providing exact results and efficient numerical methods for key physical quantities at low temperatures.

Contribution

It presents a novel scaling approach for the SK model's low-energy replica symmetry breaking solution, enabling precise calculations of physical properties.

Findings

Exact numerical value for the gap slope: 0.3014046

High-precision computations of entropy and susceptibility at low T

Efficient numerical procedure for finite timescale quantities

Abstract

We propose a simple scaling ansatz for the full replica symmetry breaking solution of the Sherrington-Kirkpatrick model in the low energy sector. This solution is shown to become exact in the limit x->0, x>>T of the Parisi replica symmetry breaking scheme parameter x. The distribution function P(x,y) of the frozen fields y has been known to develop a linear gap at zero temperature. We integrate the scaling equations to find an exact numerical value for the slope of the gap to be 0.3014046... We also use the scaling solution to devise an inexpensive numerical procedure for computing finite timescale (x=1) quantities. The entropy, the zero field cooled susceptibility and the local field distribution function are computed in the low temperature limit with high precision, barely achievable by currently available methods.