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22
Renormalization Group Pathologies and the Definition of Gibbs States
, 1998
"... We show that the socalled Renormalization Group pathologies in low temperature Ising models are due to the fact that the renormalized Hamiltonian is defined only almost everywhere (with respect to the renormalized Gibbs measures). We construct this renormalized Hamiltonian using a Renormalization G ..."
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We show that the socalled Renormalization Group pathologies in low temperature Ising models are due to the fact that the renormalized Hamiltonian is defined only almost everywhere (with respect to the renormalized Gibbs measures). We construct this renormalized Hamiltonian using a Renormalization Group method developed for random systems and we show that the pathologies are analogous to Griffiths' singularities.
Robustness of the nonGibbsian property: some examples
 J. Phys. A
, 1997
"... We discuss some examples of measures on lattice systems, which lack the property of being a Gibbs measure in a rather strong sense. 1 Introduction In recent years extensive research has been done on the occurrence of states (probability measures) on lattice systems which are not of Gibbsian type. ..."
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Cited by 24 (11 self)
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We discuss some examples of measures on lattice systems, which lack the property of being a Gibbs measure in a rather strong sense. 1 Introduction In recent years extensive research has been done on the occurrence of states (probability measures) on lattice systems which are not of Gibbsian type. Such measures occur for example in renormalizationgroup studies [17, 18, 21, 8, 9, 10, 11, 12, 13, 40], nonequilibrium statistical mechanical models [42, 26, 33, 38], image analysis [5, 15, 34], probabilistic cellular automata [25, 33] and random cluster models [19, 39]. The possibility of their occurrence and their properties have been considered by various authors [1, 7, 14, 20, 22, 24, 28, 29, 30, 31, 32, 36, 37, 41, 44]. This nonGibbsian behaviour has often been considered `pathological'  undesirable , and there have been various attempts to control the nonGibbsianness. One approach, advocated by Martinelli and Olivieri [36, 37], is to study how the nonGibbsian measures behav...
On the possible failure of the Gibbs property for measures on lattice systems.
 Markov Proc. Rel. Fields
, 1996
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Absence of Renormalization Group Pathologies Near the Critical Temperature  Two Examples
, 1996
"... : We consider real space renormalization group transformations for Ising type systems which are formally defined by e \GammaH 0 (oe 0 ) = X oe T (oe; oe 0 )e \GammaH(oe) where T (oe; oe 0 ) is a probability kernel, i.e., P oe 0 T (oe; oe 0 ) = 1, for every configuration oe. For e ..."
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Cited by 19 (1 self)
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: We consider real space renormalization group transformations for Ising type systems which are formally defined by e \GammaH 0 (oe 0 ) = X oe T (oe; oe 0 )e \GammaH(oe) where T (oe; oe 0 ) is a probability kernel, i.e., P oe 0 T (oe; oe 0 ) = 1, for every configuration oe. For each choice of the block spin configuration oe 0 , let ¯ oe 0 be the measure on spin configurations oe which is formally given by taking the probability of oe to be proportional to T (oe; oe 0 )e \GammaH(oe) . We give a condition which is sufficient to imply that the renormalized Hamiltonian H 0 is defined. Roughly speaking, the condition is that the collection of measures ¯ oe 0 are in the high temperature phase uniformly in the block spin configuration oe 0 . The proof of this result uses methods of Olivieri and Picco. We use our theorem to prove that the first iteration of the renormalization group transformation is defined in the following two examples: decimation with spaci...
Variational Principle for Some Renormalized Measures
, 1998
"... We show that some measures suffering from the socalled Renormalization Group pathologies satisfy a variational principle and that the corresponding limit of the pressure, with boundary conditions in a set of measure 1, is proportional to the pressure of the Ising model. ..."
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We show that some measures suffering from the socalled Renormalization Group pathologies satisfy a variational principle and that the corresponding limit of the pressure, with boundary conditions in a set of measure 1, is proportional to the pressure of the Ising model.
The renormalizationgroup peculiarities of Griffiths and Pearce: What have we learned?,
 in Mathematical Results in Statistical Mechanics (Proceedings of the colloquium with the same name, MarseilleLuminy,
, 1998
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Quasilocality of Projected Gibbs Measures through Analyticity Techniques
 Helv. Phys. Acta
, 1995
"... . We present two examples of projections of Gibbs measures. In the first example we prove hightemperature complete analyticity for the qstate Potts model. As a consequence we obtain complete analyticity also for the decimated Potts model. In the second example we prove quasilocality of the project ..."
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Cited by 10 (2 self)
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. We present two examples of projections of Gibbs measures. In the first example we prove hightemperature complete analyticity for the qstate Potts model. As a consequence we obtain complete analyticity also for the decimated Potts model. In the second example we prove quasilocality of the projections to the line of the pure phases of the two dimensional standard Ising system in the whole uniqueness region, and indicate why nonGibbsianness can be expected to occur for higher dimensions. 1 Introduction Over the last few years there has been a revival of interest in the mathematical welldefinedness of real space renormalization group transformations applied to lattice spin systems. This problem originally dates back to the late `70s when Griffiths and Pearce, and Israel noticed that in certain renormalization examples one cannot take for granted the existence of an effective potential [22, 23, 31]. Later a comprehensive investigation of these so called renormalizationpathologies has...