Gibbs–Bogoliubov Inequality on the Nishimori Line

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