On Testing for Parameters in Ising Models

TL;DR

This paper establishes optimal bounds for testing parameters in Ising models across various regimes and graph structures, utilizing correlation decay properties and FK-percolation theory.

Contribution

It provides the first sharp minimax separation rates for parameter testing in high temperature Ising models, including diverse graph types and regimes.

Findings

Sharp minimax separation rates derived for high temperature regimes.

Adaptive testing procedures over dependence graph and temperature.

Lower bounds for estimation and testing in two-parameter Ising models.

Abstract

We consider testing for the parameters of Ferromagnetic Ising models. While testing for the presence of possibly sparse magnetizations, we provide a general lower bound of minimax separation rates which yields sharp results in high temperature regimes. Our matching upper bounds are adaptive over both underlying dependence graph and temperature parameter. Moreover our results include the nearest neighbor model on lattices, the sparse Erd\"{o}s-R\'{e}nyi random graphs, and regular rooted trees -- right up to the critical parameter in the high temperature regime. We also provide parallel results for the entire low temperature regime in nearest neighbor model on lattices -- however in the plus boundary pure phase. Our results for the nearest neighbor model crucially depends on finite volume analogues of correlation decay property for both high and low temperature regimes -- the derivation of which borrows crucial ideas from FK-percolation theory and might be of independent interest. Finally, we also derive lower bounds for estimation and testing rates in two parameter Ising models -- which turn out to be optimal according to several recent results in this area.