Replica bound for Ising spin glass models in one dimension

TL;DR

This paper demonstrates that the interpolation method can rigorously establish lower bounds on the free energy of one-dimensional Ising spin glass models, confirming the validity of the replica symmetric cavity approach in this setting.

Contribution

The study applies the interpolation method to one-dimensional Ising spin glasses, providing rigorous bounds and validating the replica symmetric cavity method in this context.

Findings

Interpolation method yields rigorous lower bounds on free energy.

Replica symmetric cavity method is confirmed to be rigorous in 1D models.

Results hold at any finite temperature in the thermodynamic limit.

Abstract

The interpolation method is a powerful tool for rigorous analysis of mean-field spin glass models, both with and without dilution. In this study, we show that the interpolation method can be applied to Ising spin glass models in one dimension, such as a one-dimensional chain and a two-leg ladder. In one dimension, the replica symmetric (RS) cavity method is naturally expected to be rigorous for Ising spin glass models. Using the interpolation method, we rigorously prove that the RS cavity method provides lower bounds on the quenched free energies of Ising spin glass models in one dimension at any finite temperature in the thermodynamic limit.