The Posterior metric and the Goodness
of Gibbsianness
for transforms of Gibbs measures
Christof Külske, University of Groningen
Alex A Opoku, University of Groningen
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.
Assuming no a priori metric on the local state spaces but only a measurable
structure, 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.
In another application we prove the preservation of Gibbsianness for
sufficiently fine local coarse-grainings when the Hamiltonian satisfies a Lipschitz
property
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