Concentration via metastable mixing, with applications to the supercritical exponential random graph model

Folklore belief holds that metastable wells in low-temperature statistical mechanics models exhibit high-temperature behavior. We make this rigorous in the exponential random graph model (ERGM) through the lens of concentration of measure. We make use of the supercritical (low-temperature) metastable mixing which was recently proven by Bresler, Nagaraj, and Nichani, and obtain a novel concentration inequality for Lipschitz observables of the ERGM in a large metastable well, answering a question posed by those authors. To achieve this, we prove a new connectivity property for metastable mixing in the ERGM and introduce a new general result yielding concentration inequalities, which extends a result of Chatterjee. We also use a result of Barbour, Brightwell, and Luczak to cover all cases of interest. Our work extends a result of Ganguly and Nam from the subcritical (high-temperature) regime to metastable wells, and we also extend applications of this concentration, namely a central limit theorem for small subcollections of edges and a bound on the Wasserstein distance between the ERGM and the Erd\H{o}s-R\'enyi random graph. Finally, to supplement the mathematical content of the article, we present a simulation study of metastable wells in the supercritical ERGM.