The Posterior metric and the Goodness of Gibbsianness for transforms of Gibbs measures

Abstract

We present a general method to derive continuity estimates for conditional probabilities of general (possibly continuous) spin models subjected to local transformations. Such systems arise in the study of a stochastic time-evolution of Gibbs measures or as noisy observations.We exhibit the minimal necessary structure for such double-layer systems. Assuming no a priori metric on the local state spaces, we define the posterior metric on the local image space. We show that it allows in a natural way to divide the local part of the continuity estimates from the spatial part (which is treated by Dobrushin uniqueness here). We show in the concrete example of the time evolution of rotators on the (q − 1)-dimensional sphere how this method can be used to obtain estimates in terms of the familiar Euclidean metric.

Loss and recovery of Gibbsianness for XY models in external fields

“…Similar short-time results for more general compact spins can be found in Ref. 14 This paper is a continuation of Ref. 9.…”

Continuous spin mean-field models: Limiting kernels and Gibbs properties of local transforms

“…This is in nice analogy to the corresponding lattice results obtained in the paper in Ref. 24 using techniques based on the Dobrushin uniqueness. More can be said, however, about the transformed system and can be put in perspective with the corresponding lattice results.…”

“…As discussed in Ref. 24 we begin with a first-layer mean-field system, described by some given mean-field interaction ⌽ and the a priori measure ␣ with S as its single-site space. This system is coupled to a second system ͑second-layer system with SЈ as its single-site spin space͒ via K. In other words, we begin with two independent systems, namely, the first-layer system, described by ⌽ and ␣, and the second-layer system which is independent and identically distributed with distribution ␣Ј.…”

“…The study, as in Ref. 24 and the above, is based on investigating the continuity properties of the single-site conditional distribution ␥ 1 Ј for the transformed system. This consists in studying the infinite-volume N-limits of the following finite-volume quantity N Ј ͑d 1 ͉ V N−1 ͒, where the finite-volume transformed mean-field measures N Ј are second marginals of the joint measures N , i.e.,…”

Gibbs-Non-Gibbs Transitions via Large Deviations: Computable Examples

“…This specific choice is for the sake of explicit analytic computability but many results are true for a general class of rate functions that have a similar graph with two zeros located at −a, a and a maximum at zero. More formally, we require that the sequence of initial probability measures {μ n , n ∈ N} satisfies the large deviation principle with rate function i given by (8). Such rate functions arise naturally in the context of mean-field models with continuous spins and spinHamiltonian depending on the magnetization.…”