Title: Preservation and loss of the Gibbs property under single-site
transformations.
Abstract
Renormalization Group (RG) maps typically are a form of
coarse-graining. In a number of examples, going back at least to
Kadanoff , they have a random aspect as well. Indeed, renormalizing a
single configuration in such a case induces a probability distribution
on configurations instead of a single transformed configuration. The
question whether a renormalized system is Gibbsian, that is whether
the RG map is well-defined on the level of interactions, is known to
be sensitive to the precise parameters both of the map and of the
initial state. Here I generalise these issues to include the study of
single-site transformations. These maps have properties which are very
similar to those of RG transformations, even though space is not
rescaled, but they allow for different interpretations. On the one
hand we consider stochastic infinite-temperature time evolutions, for
which the question becomes whether an effective temperature exists in
the transient regime. On the other hand for a single spin
coarse-graining can be realized as a "fuzzification" or
discretization, and the question then becomes, for example, whether,
or under which conditions, a continuous-spin Gibbs measure can be
approximated by discrete-spin Gibbs measures. I present results for
both discrete-spin models and continuous-(n-vector)-spin models,
especially concentrating on recent work by C. K\"ulske, A. Opoku,
W.M. Ruszel and myself.