Mean-field theory is exact for Ising spin glass models with Kac potential in non-additive limit on Nishimori line

Recently, Mori (2011 Phys. Rev. E 84 031128) has conjectured that the free energy of Ising spin glass models with the Kac potential in the non-additive limit, such as the power-law potential in the non-additive regime, is exactly equal to that of the Sherrington-Kirkpatrick model in the thermodynamic limit. In this study, we prove that his conjecture is true on the Nishimori line at any temperature in any dimension. One of the key ingredients of the proof is the use of the Gibbs-Bogoliubov inequality on the Nishimori line. We also consider the case in which the probability distribution of the interaction is symmetric, where his conjecture is true at any temperature in one dimension but is an open problem in the low-temperature regime in two or more dimensions.

Automated summary

Recently, Mori (2011 Phys. Rev. E 84 031128) conjectured that the free energy of Ising spin glass models with the Kac potential in the non-additive limit is equal to that of the Sherrington-Kirkpatrick model in the thermodynamic limit. In this study, we prove his conjecture on the Nishimori line at any temperature in any dimension by using the Gibbs-Bogoliubov inequality. We also discuss the symmetric case of the probability distribution of the interaction, where his conjecture is true in one dimension but open in two or more dimensions in the low-temperature regime.