Higher order concentration for functions of weakly dependent random variables

TL;DR

This paper extends higher order concentration inequalities to functions of weakly dependent variables, including models like the Ising model and hypercube slices, under a logarithmic Sobolev inequality framework.

Contribution

It introduces a new concentration result for dependent variables satisfying a logarithmic Sobolev inequality, broadening the scope of higher order concentration inequalities.

Findings

Concentration results for the Ising model with weak interactions.

Extension of concentration inequalities to hypercube slices.

Application to permutations and exclusion processes.

Abstract

We extend recent higher order concentration results in the discrete setting to include functions of possibly dependent variables whose distribution (on the product space) satisfies a logarithmic Sobolev inequality with respect to a difference operator that arises from Gibbs sampler type dynamics. Examples of such random variables include the Ising model on a graph with n nodes with general, but weak interactions, i.e. in the Dobrushin uniqueness regime, for which we prove concentration results of homogeneous polynomials, as well as random permutations, and slices of the hypercube with dynamics given by either the Bernoulli-Laplace or the symmetric simple exclusion processes.