Ising Models on Dense Regular Graphs

TL;DR

This paper investigates the asymptotic behavior of one-parameter Ising models on dense regular graphs, revealing different limiting experiment regimes and estimator distributions across temperature regimes.

Contribution

It is the first to establish classical limits of experiments for Ising models and Markov random fields, detailing regime-specific Gaussian limits and estimator behaviors.

Findings

Gaussian limit in low temperature regime

Non-Gaussian limit at critical point

Gaussian limits for estimators in high temperature regime

Abstract

In this paper, we derive the limit of experiments for one parameter Ising models on dense regular graphs. In particular, we show that the limiting experiment is Gaussian in the low temperature regime, non Gaussian in the critical regime, and an infinite collection of Gaussians in the high temperature regime. We also derive the limiting distributions of the maximum likelihood and maximum pseudo-likelihood estimators, and study limiting power for tests of hypothesis against contiguous alternatives (whose scaling changes across the regimes). To the best of our knowledge, this is the first attempt at establishing the classical limits of experiments for Ising models (and more generally, Markov random fields).