A weakly periodic Gibbs measure for the ferromagnetic Potts model on a Cayley tree

We consider a nearest-neighbor inhomogeneous p-adic Potts (with q≥2 spin values) model on the Cayley tree of order k≥1. The inhomogeneity means that the interaction Jxy couplings depend on nearest-neighbors points x, y of the Cayley tree. We study (p-adic) Gibbs measures of the model. We show that (i) if q∉pℕ then there is unique Gibbs measure for any k≥1 and ∀ Jxy with | Jxy |< p-1/(p -1). (ii) For q∈p ℕ, p≥3 one can choose Jxy and k≥1 such that there exist at least two Gibbs measures which are translation-invariant.

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The uniqueness condition for a weakly periodic Gibbs measure for the hard-core model

Theoretical and Mathematical Physics ◽

10.1007/s11232-012-0120-8 ◽

2012 ◽

Vol 173 (1) ◽

pp. 1377-1386 ◽

Cited By ~ 6

Author(s):

U. A. Rozikov ◽

R. M. Khakimov

Keyword(s):

Gibbs Measure ◽

Hard Core ◽

Core Model ◽

Uniqueness Condition ◽

Periodic Gibbs Measure ◽

Hard Core Model ◽

Weakly Periodic Gibbs Measure

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Existence of p-adic quasi Gibbs measure for countable state Potts model on the Cayley tree

Journal of Inequalities and Applications ◽

10.1186/1029-242x-2012-104 ◽

2012 ◽

Vol 2012 (1) ◽

Cited By ~ 7

Author(s):

Farrukh Mukhamedov

Keyword(s):

Gibbs Measure ◽

Potts Model ◽

Cayley Tree ◽

Countable State

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Periodic Gibbs measures for models with uncountable set of spin values on a Cayley tree

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s021902571550006x ◽

2015 ◽

Vol 18 (01) ◽

pp. 1550006 ◽

Cited By ~ 5

Author(s):

U. A. Rozikov ◽

F. H. Haydarov

Keyword(s):

Gibbs Measure ◽

Cayley Tree ◽

Nearest Neighbor ◽

Gibbs Measures ◽

Sufficient Condition ◽

Translation Invariant ◽

Periodic Gibbs Measure

We consider models with nearest-neighbor interactions and with the set [0, 1] of spin values, on a Cayley tree of order k ≥ 1. We show that periodic Gibbs measures are either translation-invariant or periodic with period two. We describe two-periodic Gibbs measures of the model. For k = 1 we show that there is no any periodic Gibbs measure. In case k ≥ 2 we get a sufficient condition on Hamiltonian of the model with uncountable set of spin values under which the model has no periodic Gibbs measure. We construct several models which have at least two periodic Gibbs measures.

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