On inhomogeneous p-adic potts model on a cayley tree

Farrukh Mukhamedov; Utkir Rozikov

doi:10.1142/S0219025705001974

On inhomogeneous p-adic potts model on a cayley tree

Farrukh Mukhamedov
, Utkir Rozikov

Research output: Contribution to journal › Article › peer-review

54 Citations (Scopus)

Abstract

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 J_xy 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 ∀ J_xy with |J_xy| < p ^-1/(p-1). (ii) For q ∈ pℕ, p ≥ 3 one can choose J _xy and k ≥ l such that there exist at least two Gibbs measures which are translation-invariant.

Original language	English
Pages (from-to)	277-290
Number of pages	14
Journal	Infinite Dimensional Analysis, Quantum Probability and Related Topics
Volume	8
Issue number	2
DOIs	https://doi.org/10.1142/S0219025705001974
Publication status	Published - Jun 2005
Externally published	Yes

Keywords

Cayley tree
Gibbs measure
P-adic field
Potts model

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Statistics and Probability
Mathematical Physics
Applied Mathematics

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10.1142/S0219025705001974

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title = "On inhomogeneous p-adic potts model on a cayley tree",

abstract = "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 ∈ pℕ, p ≥ 3 one can choose J xy and k ≥ l such that there exist at least two Gibbs measures which are translation-invariant.",

keywords = "Cayley tree, Gibbs measure, P-adic field, Potts model",

author = "Farrukh Mukhamedov and Utkir Rozikov",

note = "Funding Information: The work was done within the scheme of Borsa di Studio NATO-CNR. F.M. would like to thank the Italian CNR for providing financial support and II Universit{\'a} di Roma “Tor Vergata” for all facilities. U.R. thanks Institute des Hautes Etudes Sci-entifiques (IHES) for supporting the visit to IHES, Bures-sur-Yvette in September-December 2003 and the IGS programme at the Isaac Newton Institute (INI) for supporting the visit to INI in November 2003. The authors also acknowledge their gratitude to Professors I. V. Volovich and A. Yu. Khrennikov for helpful comments and discussions. The work was also partially supported by Grants Φ.1.1.12 and Φ.2.1.56 of CST of the Republic Uzbekistan. The authors are grateful to the referee{\textquoteright}s helpful suggestions.",

year = "2005",

month = jun,

doi = "10.1142/S0219025705001974",

language = "English",

volume = "8",

pages = "277--290",

journal = "Infinite Dimensional Analysis, Quantum Probability and Related Topics",

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AU - Mukhamedov, Farrukh

AU - Rozikov, Utkir

N1 - Funding Information: The work was done within the scheme of Borsa di Studio NATO-CNR. F.M. would like to thank the Italian CNR for providing financial support and II Universitá di Roma “Tor Vergata” for all facilities. U.R. thanks Institute des Hautes Etudes Sci-entifiques (IHES) for supporting the visit to IHES, Bures-sur-Yvette in September-December 2003 and the IGS programme at the Isaac Newton Institute (INI) for supporting the visit to INI in November 2003. The authors also acknowledge their gratitude to Professors I. V. Volovich and A. Yu. Khrennikov for helpful comments and discussions. The work was also partially supported by Grants Φ.1.1.12 and Φ.2.1.56 of CST of the Republic Uzbekistan. The authors are grateful to the referee’s helpful suggestions.

PY - 2005/6

Y1 - 2005/6

N2 - 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 ∈ pℕ, p ≥ 3 one can choose J xy and k ≥ l such that there exist at least two Gibbs measures which are translation-invariant.

AB - 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 ∈ pℕ, p ≥ 3 one can choose J xy and k ≥ l such that there exist at least two Gibbs measures which are translation-invariant.

KW - Cayley tree

KW - Gibbs measure

KW - P-adic field

KW - Potts model

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On inhomogeneous p-adic potts model on a cayley tree

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Keywords

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