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We discuss the status of recent Gibbsian descriptions of the restriction (projection) of the Ising phases to a layer. We concentrate on the projection of the two-dimensional low temperature Ising phases for which we prove a variational principle.

. We investigate the complete analyticity (CA) of the two-dimensional q- state Potts model for large values of q. We are able to prove it for every temperature T ? Tcr (q), provided we restrict ourselves to nice subsets, their niceness depending on the temperature T . Contrary to this restricted complete analyticity (RCA), the full CA is known to fail for some values of the temperature above Tcr (q). Our proof is based on Pirogov-Sinai theory and cluster expansions for the FortuinKasteleyn representation, which are available for the Potts model at all temperatures, provided q is large enough. 1. Introduction. In this paper we are dealing with the two-dimensional q-state Potts model, which is the statistical mechanics model on Z 2 with formal Hamiltonian H(oe) = \Gamma X fx;yg ffi oe(x);oe(y) ; (1:1) where oe(x) = 1; : : : ; q is the spin variable at the site x 2 Z 2 , ffi oe(x);oe(y) is 1 for oe(x) = oe(y) and is 0 otherwise, the summation is taken over nearest neighbors, a...

We study local transformations of Gibbs measures. We establish sufficient conditions for the quasilocality of the images and obtain results on the existence and continuity properties of their relative energies. General results are illustrated by simple examples.

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We review what we have learned about the “Renormalization Group peculiarities ” which were discovered more than twentyfive years ago by Griffiths and Pearce. We discuss which of the questions they asked have been answered and which ones are still widely open. The problems they raised have led to the study of non-Gibbsian states (probability measures). We also mention some further related developments, which find applications in nonequilibrium questions and disordered models.

Renormalization transformations were developed by theoretical physicists in order to investigate first problems arising in quantum field theory and later in statistical mechanics, specifically phase transitions and critical phenomena appearing in systems of a large number of interacting components. In their latter version they provide a scheme of systematic