  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.
