Towrad mean-field bound for critical temperature on Nishimori line

The critical inverse temperature of the mean-field approximation gives a lower bound of the true critical inverse temperature in a broad class of ferromagnetic spin models, which is called the mean-field bound for the critical temperature. In this study, we explore the possibility of a corresponding mean-field bound for the critical temperature in Ising spin-glass models with Gaussian randomness on the Nishimori line. On the Nishimori line, the critical inverse temperature of the mean-field approximation is given by βMFNL=√1/z, where z denotes the coordination number. Using the Griffiths inequalities on the Nishimori line, we prove that there is zero spontaneous magnetization in the high-temperature region β < βMFNL/2. In other words, the true critical inverse temperature βcNL of the Nishimori line is always bounded by βcNL ≥ βMFNL/2. Unfortunately, we did not succeed in obtaining the corresponding mean-field bound βcNL ≥ βMFNL on the Nishimori line.