Parisi Formula for Balanced Potts Spin Glass

Abstract

The Potts spin glass is a generalization of the Sherrington–Kirkpatrick (SK) model that allows for spins to take more than two values. Based on a novel synchronization mechanism, Panchenko (Ann Probab 46(2):829–864, 2018) showed that the limiting free energy is given by a Parisi-type variational formula. The functional order parameter in this formula is a probability measure on a monotone path in the space of positive-semidefinite matrices. By comparison, the order parameter for the SK model is much simpler: a probability measure on the unit interval. Nevertheless, a longstanding prediction by Elderfield and Sherrington (J Phys C Solid State Phys 16(15):L497–L503, 1983) is that the order parameter for the Potts spin glass can be reduced to that of the SK model. We prove this prediction for the balanced Potts spin glass, where the model is constrained so that the fraction of spins taking each value is asymptotically the same. It is generally believed that the limiting free energy of the balanced model is the same as that of the unconstrained model, in which case our results reduce the functional order parameter of Panchenko’s variational formula to probability measures on the unit interval. The intuitive reason—for both this belief and the Elderfield–Sherrington prediction—is that no spin value is a priori preferred over another, and the order parameter should reflect this inherent symmetry. This paper rigorously demonstrates how symmetry, when combined with synchronization, acts as the desired reduction mechanism. Our proof requires that we introduce a generalized Potts spin glass model with mixed higher-order interactions, which is interesting it its own right. We prove that the Parisi formula for this model is differentiable with respect to inverse temperatures. This is a key ingredient for guaranteeing the Ghirlanda–Guerra identities without perturbation, which then allow us to exploit symmetry and synchronization simultaneously.