Concentration Inequalities for Functions of Gibbs Fields with Application to Diffraction and Random Gibbs Measures

Abstract

We derive useful general concentration inequalities for functions of Gibbs fields in the uniqueness regime. We also consider expectations of random Gibbs measures that depend on an additional disorder field, and prove concentration w.r.t. the disorder field. Both fields are assumed to be in the uniqueness regime, allowing in particular for non-independent disorder fields. The modification of the bounds compared to the case of an independent field can be expressed in terms of constants that resemble the Dobrushin contraction coefficient, and are explicitly computable. On the basis of these inequalities, we obtain bounds on the deviation of a diffraction pattern created by random scatterers located on a general discrete point set in Euclidean space, restricted to a finite volume. Here we also allow for thermal dislocations of the scatterers around their equilibrium positions. Extending recent results for independent scatterers, we give a universal upper bound on the probability of a deviation of the random scattering measures applied to an observable from its mean. The bound is exponential in the number of scatterers with a rate that involves only the minimal distance between points in the point set.