Mean Field Spin Glass Models Under Weak External Field

Abstract

We study the fluctuation and limiting distribution of free energy in mean-field Ising spin glass models under weak external fields. We prove that at high temperature, there are three sub-regimes concerning the strength of external field $h \approx \rho N^{-\alpha }$ with $\rho ,\alpha \in (0,\infty )$. In the super-critical regime $\alpha < 1/4$, the variance of the log-partition function is $\approx N^{1-4\alpha }$. In the critical regime $\alpha = 1/4$, the fluctuation is of constant order but depends on $\rho$. Whereas, in the sub-critical regime $\alpha >1/4$, the variance is $\Theta (1)$ and does not depend on $\rho$. We explicitly express the asymptotic mean and variance in all three regimes and prove Gaussian central limit theorems. Our proofs mainly follow two approaches. One generalizes quadratic coupling and Guerra’s interpolation scheme for Gaussian disorder, extending to other spin glass models. This approach can establish CLT at high temperature by proving a limiting variance result for the overlap. The other combines cluster expansion for general symmetric disorders and multivariate Stein’s method for exchangeable pairs. For the zero external field case, cluster expansion was first used in the seminal work of Aizenman et al. (Commun Math Phys 112(1):3–20, 1987). This approach is believed not to work if the external field is present. We show that if the external field is present but not too strong, it still works with a new cluster structure. In particular, we prove the CLT up to the critical temperature in the Sherrington–Kirkpatrick (SK) model when $\alpha \geqslant 1/4$. We further address the generality of this cluster-based approach. Specifically, we give limiting results for the multi-species and diluted SK models. We believe this approach will shed new light on how the external field affects the general spin glass system. We also obtain explicit convergence rates when applying Stein’s method to establish the CLTs.