Spin glass models with quadratic-type Hamiltonians are disordered statistical physics systems with competing ferromagnetic and anti-ferromagnetic spin interactions. The corresponding Gibbs measures belong to the exponential family parametrized by (inverse) temperature $\beta>0$ and external field $h\in\mathbb{R}$. Given a sample from these Gibbs measures, a statistically fundamental question is to infer the temperature and external field parameters. In 2007, Chatterjee (Ann. Statist. 35 (2007), no.5, 1931-1946) first proved that in the absence of external field $h=0$, the maximum pseudolikelihood estimator for $\beta$ is $\sqrt{N}$-consistent under some mild assumptions on the disorder matrices. It was left open whether the same method can be used to estimate the temperature and external field simultaneously. In this paper, under some easily verifiable conditions, we prove that the bivariate maximum pseudolikelihood estimator is indeed jointly $\sqrt{N}$-consistent for the temperature and external field parameters. The examples cover the classical Sherrington-Kirkpatrick model and its diluted variants.

翻译：具有二次型哈密顿量的自旋玻璃模型是包含铁磁与反铁磁自旋相互竞争的无序统计物理系统。相应的吉布斯测度属于以（逆）温度 $\beta>0$ 和外场 $h\in\mathbb{R}$ 为参数的指数族。给定来自这些吉布斯测度的一个样本，一个统计学基本问题是如何推断温度和外场参数。2007年，Chatterjee (Ann. Statist. 35 (2007), no.5, 1931-1946) 首次证明了在外场 $h=0$ 的情况下，在无序矩阵满足某些温和假设下，$\beta$ 的最大伪似然估计量具有 $\sqrt{N}$ 相合性。但该方法能否同时用于估计温度和外场的问题尚未解决。本文证明，在一些易于验证的条件下，二元最大伪似然估计量对温度和外场参数确实具有联合 $\sqrt{N}$ 相合性。所涵盖的示例包括经典的 Sherrington-Kirkpatrick 模型及其稀释变体。