Griffiths Inequalities in the Nishimori Line

The Griths inequalities for Ising spin-glass models with Gaussian randomness of non- vanishing mean are proved using the properties of the Gaussian distribution and the gauge symmetry of the system. These inequalities imply that the correlation functions are non- negative and monotonic along the Nishimori line in the phase diagram. From this result, the existence of the thermodynamic limit for the correlation functions and the free energy is proved under free and xed boundary conditions. Relations between the location of multi- critical points are also derived for dieren t lattices. x1. Introduction The Griths inequalities give us signican t knowledge about phase transitions in ferromagnetic Ising models.1),2) These inequalities are composed of two state- ments: correlation functions are non-negative and increase monotonically with the interaction among any set of the spins. From the Griths inequalities, the existence of the free energy per spin and correlation functions is proved under several boundary conditions. Furthermore, relations on critical points for various lattices are derived. However, since the proof needs the conditions that all the interactions are ferromag- netic, there was no proof of similar inequalities for the spin-glass models which have both ferromagnetic and antiferromagnetic interactions with non-vanishing mean. Recently, the rst Griths inequality has been proved for the Sherrington- Kirkpatrick model and the Edward-Anderson model using integration by parts.3),4) Moreover, in 3) and 4), monotonicity of correlation functions is proved not with respect to the strength of the interaction but with the variance of the randomness. On the other hand, the gauge theory, which uses gauge symmetry of the system, is known to be useful for analytic investigations in spin-glass models, yielding various exact results on a line called the Nishimori line (NL).5),6) We have been able to prove both Griths inequalities with respect to the mean of the randomness for the Gaussian spin glass on the NL using the gauge theory and the technique of integra- tion by parts.7) The resulting inequalities can be used to prove the existence of the thermodynamic limit for correlation functions and the free energy and to derive inequalities on the location of the multicritical point for various lattices. The present contribution briey reviews these results. In the next section, we present our results and outline their proof. Applications of these inequalities are discussed in the third section.