Gibbs-Bogoliubov inequality on Nishimori line

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.

9 pages, 0 figure

References in corpus (2)

• Variational Principle of Bogoliubov and Generalized Mean Fields in Many-Particle Interacting Systems
• The solution of the deep Boltzmann machine on the Nishimori line

Cited by in corpus (5)

• Anomalous distribution of magnetization in an Ising spin glass with correlated disorder
• Mean-field theory is exact for Ising spin glass models with Kac potential in non-additive limit on Nishimori line
• Gauge theory for mixed $p$-spin glasses
• Gauge theory for quantum XYZ spin glasses
• Toward mean-field bound for critical temperature on Nishimori line