Large deviations and the maximum entropy principle for marked point random fields

Abstract

Summary.We establish large deviation principles for the stationary and the individual empirical fields of Poisson, and certain interacting, random fields of marked point particles in IR~. The underlying topologies are induced by a class of not necessarily bounded local functions, and thus finer than the usual weak topologies. Our methods yield further that the limiting behaviour of conditional Poisson distributions, as well as certain distributions of Gibbsian type, is governed by the maximum entropy principle. We also discuss various applications and examples.

Large deviations and the equivalence of ensembles for Gibbsian particle systems with superstable interaction

“…[10]. In this paper we study large deviations from the ergodic theorem (1.2) when P is Gibbsian relative to a suitable pair interaction q~.…”

“…A configuration of particles (without multiple occupancies) is de-scribed by a locally finite subset co of R d, i.e., a set co C R d having finite intersection with every bounded set. We write f2 for the set of all such configurations co. ~2 is equipped with the a-algebra 9 c generated by the counting variables Ne:co--+ card(co N B) for Bore1 subsets B of R d. It is well-known [10,13] that 5 c is the Borel a-algebra for a natural Polish topology on Q. The translation group 0 = (OX)xERd acting on (~,S) is defined by 0xco = {y -x :y E co},co c f~,x c R a.…”

The existence of quermass-interaction processes for nonlocally stable interaction and nonbounded convex grains

“…The following lemma, for which the proof is almost the same as the one given in [4], ensures that (μ n ) and (μ n ) are asymptotically equivalent for the topology τ W .…”

“…Let us remark that Assumption 2.1 of [4] with the function ψ : R → 1 + R 2 is exactly our assumption (H). So, it plays a fundamental role at this part of the proof of Theorem 2.1.…”

“…The topology used here is the weak* topology induced by the local and tame functions on M. Georgii and Zessin [4] proved that a set of measures is relatively compact if and only if the specific entropy of these measures is uniformly bounded (see Proposition 3.2, below). Therefore, we show via the fundamental Proposition 3.3, below, that the specific entropy of the measures (μ n ) is uniformly bounded and we obtain the convergence of a subsequence (μ ϕ(n) ) to a measureμ.…”