A note on concentration for polynomials in the Ising model

TL;DR

This paper establishes precise exponential concentration inequalities for polynomials in Ising models under the Dobrushin condition, extending Gaussian tail estimates and Hanson-Wright inequalities to dependent variables.

Contribution

It introduces multilevel exponential concentration bounds for polynomials in Ising models, including quadratic forms, under the Dobrushin condition, generalizing classical results for independent variables.

Findings

Derived multilevel exponential concentration inequalities for polynomials in Ising models.

Extended Hanson-Wright inequality to dependent Ising model variables.

Provided concentration results for convex functions and quadratic forms in dependent settings.

Abstract

We present precise multilevel exponential concentration inequalities for polynomials in Ising models satisfying the Dobrushin condition. The estimates have the same form as two-sided tail estimates for polynomials in Gaussian variables due to Lata{\l}a. In particular, for quadratic forms we obtain a Hanson-Wright type inequality. We also prove concentration results for convex functions and estimates for nonnegative definite quadratic forms, analogous as for quadratic forms in i.i.d. Rademacher variables, for more general random vectors satisfying the approximate tensorization property for entropy.