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 sites 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.

We derive multi-level concentration inequalities for polynomials in independent random variables with an α-sub-exponential tail decay. A particularly interesting case is given by quadratic forms f (X1, . . . , Xn) = X, AX , for which we prove Hanson-Wright-type inequalities with explicit dependence on various norms of the matrix A. A consequence of these inequalities is a two-level concentration inequality for quadratic forms in α-sub-exponential random variables, such as quadratic Poisson chaos.We provide various applications of these inequalities. Among them are generalizations of some results proven by Rudelson and Vershynin from sub-Gaussian to α-sub-exponential random variables, i. e. concentration of the Euclidean norm of the linear image of a random vector and concentration inequalities for the distance between a random vector and a fixed subspace.

We study sharpened forms of the concentration of measure phenomenon typically centered at stochastic expansions of order d − 1 for any d ∈ N. The bounds are based on d-th order derivatives or difference operators. In particular, we consider deviations of functions of independent random variables and differentiable functions over probability measures satisfying a logarithmic Sobolev inequality, and functions on the unit sphere. Applications include concentration inequalities for U -statistics as well as for classes of symmetric functions via polynomial approximations on the sphere (Edgeworth-type expansions).

We derive sufficient conditions for a probability measure on a finite product space (a spin system) to satisfy a (modified) logarithmic Sobolev inequality. We establish these conditions for various examples, such as the (vertex-weighted) exponential random graph model, the random coloring and the hard-core model with fugacity.This leads to two separate branches of applications. The first branch is given by mixing time estimates of the Glauber dynamics. The proofs do not rely on coupling arguments, but instead use functional inequalities. As a byproduct, this also yields exponential decay of the relative entropy along the Glauber semigroup. Secondly, we investigate the concentration of measure phenomenon (particularly of higher order) for these spin systems. We show the effect of better concentration properties by centering not around the mean, but a stochastic term in the exponential random graph model. From there, one can deduce a central limit theorem for the number of triangles from the CLT of the edge count. In the Erdös-Rényi model the first order approximation leads to a quantification and a proof of a central limit theorem for subgraph counts.

In this paper, we prove multilevel concentration inequalities for bounded functionals f = f (X 1 , . . . , X n ) of random variables X 1 , . . . , X n that are either independent or satisfy certain logarithmic Sobolev inequalities. The constants in the tail estimates depend on the operator norms of k-tensors of higher order differences of f . We provide applications for both dependent and independent random variables. This includes deviation inequalities for empirical processes f (X ) = sup g∈F |g(X )| and suprema of homogeneous chaos in bounded random variables in the Banach space caseThe latter application is comparable to earlier results of Boucheron, Bousquet, Lugosi, and Massart and provides the upper tail bounds of Talagrand. In the case of Rademacher random variables, we give an interpretation of the results in terms of quantities familiar in Boolean analysis. Further applications are concentration inequalities for U -statistics with bounded kernels h and for the number of triangles in an exponential random graph model. Keywords Concentration of measure • Empirical processes • Functional inequalities • Hamming cube • Logarithmic Sobolev inequality • Product spaces • Suprema of chaos • Weakly dependent random variables Mathematics Subject Classification (2010) 60E15 • 05C80 This research was supported by the German Research Foundation (DFG) via CRC 1283 "Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications".