# Concentration inequalities for polynomials of contracting Ising models

## TL;DR

This paper establishes concentration inequalities for degree-$d$ polynomials of spins in Ising models under contracting Glauber dynamics, extending known results for linear and quadratic cases to higher degrees.

## Contribution

It provides new concentration bounds for polynomials of arbitrary fixed degree in Ising models with contracting dynamics, generalizing previous results for lower degrees.

## Key findings

- Variance of the polynomial is $O(N^d)$.
- Tail probabilities decay as $ ext{exp}(- c r^{2/d})$ for large deviations.
- Results hold for fixed degree $d$ with $O(1)$ coefficients.

## Abstract

We study the concentration of a degree-$d$ polynomial of the $N$ spins of a general Ising model, in the regime where single-site Glauber dynamics is contracting. For $d=1$, Gaussian concentration was shown by Marton (1996) and Samson (2000) as a special case of concentration for convex Lipschitz functions, and extended to a variety of related settings by e.g., Chazottes et al. (2007) and Kontorovich and Ramanan (2008). For $d=2$, exponential concentration was shown by Marton (2003) on lattices. We treat a general fixed degree $d$ with $O(1)$ coefficients, and show that the polynomial has variance $O(N^d)$ and, after rescaling it by $N^{-d/2}$, its tail probabilities decay as $\exp(- c\, r^{2/d})$ for deviations of $r \geq C \log N$.