- 02-251 Christof Kuelske
- Concentration inequalities for functions of Gibbs fields with application to diffraction
and random Gibbs measures
Jun 2, 02
(auto. generated ps),
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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 field.
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 the 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
an upper bound rate that involves only the minimal
distance between points in the point set.