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19
Absence of phase transition for antiferromagnetic Potts models via the Dobrushin uniqueness theorem
- J. Stat. Phys
, 1997
"... Dedicated to the memory of R.L. Dobrushin We prove that the q-state Potts antiferromagnet on a lattice of maximum coordination number r exhibits exponential decay of correlations uniformly at all temperatures (including zero temperature) whenever q> 2r. We also prove slightly better bounds for se ..."
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Cited by 67 (4 self)
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Dedicated to the memory of R.L. Dobrushin We prove that the q-state Potts antiferromagnet on a lattice of maximum coordination number r exhibits exponential decay of correlations uniformly at all temperatures (including zero temperature) whenever q> 2r. We also prove slightly better bounds for several two-dimensional lattices: square lattice (exponential decay for q ≥ 7), triangular lattice (q ≥ 11), hexagonal lattice (q ≥ 4), and Kagomé lattice (q ≥ 6). The proofs are based on the Dobrushin uniqueness theorem. Key Words: Dobrushin uniqueness theorem, antiferromagnetic Potts models, phase transition. 1
Renormalization Group Pathologies and the Definition of Gibbs States
, 1998
"... We show that the so-called 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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Cited by 33 (0 self)
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We show that the so-called 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 non-Gibbsian 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 renormalization-group studies [17, 18, 21, 8, 9, 10, 11, 12, 13, 40], non-equilibrium 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 non-Gibbsian 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 non-Gibbsian measures behav...
On the possible failure of the Gibbs property for measures on lattice systems.
- Markov Proc. Rel. Fields
, 1996
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The renormalization-group peculiarities of Griffiths and Pearce: What have we learned?,
- in Mathematical Results in Statistical Mechanics (Proceedings of the colloquium with the same name, Marseille-Luminy,
, 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 high-temperature complete analyticity for the q-state 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 high-temperature complete analyticity for the q-state 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 non-Gibbsianness 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 renormalization-pathologies has...
