Non-uniqueness of Gibbs Measure for Models with Uncountable Set of Spin Values on a Cayley Tree

Èshkabilov, Yu. Kh.; Haydarov, F. H.; Rozikov, U. A.

doi:10.1007/s10955-012-0494-x

Cited by 25 publications

(1 citation statement)

References 13 publications

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“…For a lattice model with a compact set of spin values, it is known that the set of Gibbs measures is non-empty ( [3], [4], [5], [6]). But for models with non-compact set of spin values, the existence problem of the Gibbs measure remains one of the important problems in physics and statistical mechanics.…”

Section: Introductionmentioning

confidence: 99%

Mirror symmetry of height-periodic gradient Gibbs measures of a SOS model on Cayley trees

Rozikov

2022

Preprint

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For the solid-on-solid (SOS) model with spin values from the set of all integers on a Cayley tree we give gradient Gibbs measures (GGMs). Such a measure corresponds to a boundary law (which is an infinite-dimensional vector-valued function defined on vertices of the Cayley tree) satisfying an infinite system of functional equations. We give several concrete GGMs of boundary laws which are independent from vertices of the Cayley tree and (as an infinite-dimensional vector) have periodic, (non-)mirror symmetric coordinates.

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Section: Introductionmentioning

confidence: 99%

Mirror symmetry of height-periodic gradient Gibbs measures of a SOS model on Cayley trees

Rozikov

2022

Preprint

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Mirror Symmetry of Height-Periodic Gradient Gibbs Measures of an SOS Model on Cayley Trees

Rozikov

2022

J Stat Phys

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Uniqueness of Gibbs Measure for Models with Uncountable Set of Spin Values on a Cayley Tree

Èshkabilov

Haydarov

Rozikov³

2012

Math Phys Anal Geom

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We consider models with nearest-neighbor interactions and with the set [0, 1] of spin values, on a Cayley tree of order k ≥ 1. It is known that the "splitting Gibbs measures" of the model can be described by solutions of a nonlinear integral equation. For arbitrary k ≥ 2 we find a sufficient condition under which the integral equation has unique solution, hence under the condition the corresponding model has unique splitting Gibbs measure.Mathematics Subject Classifications (2010). 82B05, 82B20 (primary); 60K35 (secondary)

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Non-uniqueness of Gibbs Measure for Models with Uncountable Set of Spin Values on a Cayley Tree

Cited by 25 publications

References 13 publications

Mirror symmetry of height-periodic gradient Gibbs measures of a SOS model on Cayley trees

Mirror symmetry of height-periodic gradient Gibbs measures of a SOS model on Cayley trees

Mirror Symmetry of Height-Periodic Gradient Gibbs Measures of an SOS Model on Cayley Trees

Uniqueness of Gibbs Measure for Models with Uncountable Set of Spin Values on a Cayley Tree

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