Gibbs-Bogoliubov inequality on Nishimori line

TL;DR

This paper extends the Gibbs-Bogoliubov inequality to Nishimori line in Ising spin-glass models, showing the quenched free energy is bounded by a trial function, with implications for mean-field models like SK and p-spin.

Contribution

It introduces a novel inequality on the Nishimori line for spin-glass models, revealing that the bound matches the replica-symmetric solution in mean-field cases.

Findings

The inequality holds on the Nishimori line for Gaussian spin glasses.

The bound coincides with the replica-symmetric solution in mean-field models.

Convexity of the pressure function is key to the proof.

Abstract

The Gibbs-Bogoliubov inequality states that the free energy of a system is always lower than that calculated by a trial function. In this study, we show that a counterpart of the Gibbs-Bogoliubov inequality holds on the Nishimori line for Ising spin-glass models with Gaussian randomness. Our inequality states that the quenched free energy of a system is always lower than that calculated using a quenched trial function. The key component of the proof is the convexity of the pressure function $\mathbb{E}\left[\log Z_{} \right]$ with respect to the parameters along the Nishimori line, which differs from the conventional convexity with respect to the inverse temperature. When our inequality was applied to mean-field models, such as the Sherrington-Kirkpatrick model and $p$-spin model, the bound coincided with the replica-symmetric solution indicating that the equality holds.