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Gibbsian properties and convergence of the iterates for the Block-Averaging Transformation
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Transformations of Gibbs measures
, 1998
"... 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 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.
Renormalization Group, Non-Gibbsian states, their relationship and further developments
, 2005
"... 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 ..."
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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.
Toward a mathematical theory of renormalization
"... 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. ..."
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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
Effect of self–interaction on the phase diagram of a Gibbs–like mea- sure derived by a reversible Probabilistic Cellular Automata
"... lands Abstract. Cellular Automata are discrete–time dynamical systems on a spatially extended discrete space which provide paradigmatic examples of nonlinear phenomena. Their stochas-tic generalizations, i.e., Probabilistic Cellular Automata (PCA), are discrete time Markov chains on lattice with fin ..."
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lands Abstract. Cellular Automata are discrete–time dynamical systems on a spatially extended discrete space which provide paradigmatic examples of nonlinear phenomena. Their stochas-tic generalizations, i.e., Probabilistic Cellular Automata (PCA), are discrete time Markov chains on lattice with finite single–cell states whose distinguishing feature is the parallel character of the updating rule. We study the ground states of the Hamiltonian and the low–temperature phase diagram of the related Gibbs measure naturally associated with a class of reversible PCA, called the cross PCA. In such a model the updating rule of a cell depends indeed only on the status of the five cells forming a cross centered at the original cell itself. In particular, it depends on the value of the center spin (self–interaction). The goal of the paper is that of investigating the role played by the self–interaction parameter in connection with the ground states of the Hamiltonian and the low–temperature phase diagram of the Gibbs measure associated with this particular PCA. Pacs: 05.45.-a; 05.50.+q; 64.60.De
