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Almost Gibbsian versus weakly Gibbsian measures
 STOCH. PROC. APPL
, 1999
"... We consider two possible extensions of the standard de nition of Gibbs measures for lattice spin systems. When a random eld has conditional distributions which are almost surely continuous (almost Gibbsian eld), then there is a potential for that eld which is almost surely summable (weakly Gibbsian ..."
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Cited by 37 (13 self)
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We consider two possible extensions of the standard de nition of Gibbs measures for lattice spin systems. When a random eld has conditional distributions which are almost surely continuous (almost Gibbsian eld), then there is a potential for that eld which is almost surely summable (weakly Gibbsian eld). This generalizes the standard Kozlov theorems. The converse is not true in general as is illustrated by counterexamples.
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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Cited by 11 (1 self)
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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 Restriction of the Ising Model to a Layer
, 1998
"... 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 twodimensional low temperature Ising phases for which we prove a variational principle. ..."
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Cited by 8 (7 self)
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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 twodimensional low temperature Ising phases for which we prove a variational principle.
Freezing transition in the Ising model without internal contours. Probab. Theory Related Fields
, 1999
"... Abstract. We consider the low temperature Ising model in a uniform magnetic field h>0 with minus boundary conditions and conditioned on having no internal contours. This simple contour model defines a nonGibbsian spin state. For large enough magnetic fields (h>hc) this state is concentrated o ..."
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Cited by 3 (0 self)
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Abstract. We consider the low temperature Ising model in a uniform magnetic field h>0 with minus boundary conditions and conditioned on having no internal contours. This simple contour model defines a nonGibbsian spin state. For large enough magnetic fields (h>hc) this state is concentrated on the single spin configuration of all spins up. For smaller values (h ≤ hc), the spin state is nontrivial. At the critical point hc = 0 the magnetization jumps discontinuously. Freezing provides also an example of a translation invariant weakly Gibbsian state which is not almost Gibbsian. Mathematical Subject Classification (1991): 60G, 82B20, 82B31, 82B26 1.
Problems with the definition of renormalized Hamiltonians for momentumspace renormalization transformations
, 1998
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A Cluster Expansion Approach to the Renormalization Group Transformations, Preprint on arXiv
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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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Cited by 1 (0 self)
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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.
[1] NonGibbsianness of the invariant measures of nonreversible cellular automata with totally asymmetric noise
, 2001
"... We present a class of random cellular automata with multiple invariant measures which are all d nonGibbsian. The automata have configuration space {0,1} Z, with d> 1, and they are noisy versions of automata with the “eroder property”. The noise is totally asymmetric in the sense that it allows r ..."
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We present a class of random cellular automata with multiple invariant measures which are all d nonGibbsian. The automata have configuration space {0,1} Z, with d> 1, and they are noisy versions of automata with the “eroder property”. The noise is totally asymmetric in the sense that it allows random flippings of “0 ” into “1 ” but not the converse. We prove that all invariant measures assign to the event “a sphere with a large radius L is filled with ones ” a probability µL that is too large for the measure to be Gibbsian. For example, for the NEC automaton ( − ln µL) ≍ L while for any Gibbs measure the corresponding value is ≍ L2. Key words: Gibbs vs. nonGibbs measures, cellular automata, invariant measures, nonergodicity, eroders, convex sets.
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