On the uniqueness of Gibbs measures for p-adic nonhomogeneous λ-model on the Cayley tree

Murod Khamraev; Farruh Mukhamedov; Utkir Rozikov

doi:10.1007/s11005-004-3500-7

On the uniqueness of Gibbs measures for p-adic nonhomogeneous λ-model on the Cayley tree

Murod Khamraev, Farruh Mukhamedov, Utkir Rozikov

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

18 Citations (Scopus)

Abstract

We consider a nearest-neighbor p-adic λ-model with spin values ±1 on a Cayley tree of order k ≥ 1. We prove for the model there is no phase transition and as well as being unique, the p-adic Gibbs measure is bounded if and only if p ≥ 3. If p = 2, then we find a condition which guarantees the nonexistence of a phase transition. Besides, the results are applied to the p-adic Ising model and we show that for the model there is a unique p-adic Gibbs measure.

Original language	English
Pages (from-to)	17-28
Number of pages	12
Journal	Letters in Mathematical Physics
Volume	70
Issue number	1
DOIs	https://doi.org/10.1007/s11005-004-3500-7
Publication status	Published - Oct 2004
Externally published	Yes

Keywords

Cayley tree
Gibbs measure
Ising model
p-adic field
λ-model

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics

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10.1007/s11005-004-3500-7

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title = "On the uniqueness of Gibbs measures for p-adic nonhomogeneous λ-model on the Cayley tree",

abstract = "We consider a nearest-neighbor p-adic λ-model with spin values ±1 on a Cayley tree of order k ≥ 1. We prove for the model there is no phase transition and as well as being unique, the p-adic Gibbs measure is bounded if and only if p ≥ 3. If p = 2, then we find a condition which guarantees the nonexistence of a phase transition. Besides, the results are applied to the p-adic Ising model and we show that for the model there is a unique p-adic Gibbs measure.",

keywords = "Cayley tree, Gibbs measure, Ising model, p-adic field, λ-model",

author = "Murod Khamraev and Farruh Mukhamedov and Utkir Rozikov",

note = "Funding Information: F.M. thanks the Italian NATO-CNR (2003) for providing financial support and II Universita di Roma {\textquoteleft}Tor Vergata{\textquoteright} for all the facilities they provided. U.R. thanks Institute des Hautes Etudes Scientifiques (IHES) for supporting the visit to Bures-sur-Yvette (IHES, France) in September–December 2003. The authors also acknowledge with gratitude the helpful comments and discussions of Professors I. V. Volovich and A. Yu. Khrennikov. The authors also express their gratitude to the referee for some useful observations.",

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doi = "10.1007/s11005-004-3500-7",

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AU - Khamraev, Murod

AU - Mukhamedov, Farruh

AU - Rozikov, Utkir

N1 - Funding Information: F.M. thanks the Italian NATO-CNR (2003) for providing financial support and II Universita di Roma ‘Tor Vergata’ for all the facilities they provided. U.R. thanks Institute des Hautes Etudes Scientifiques (IHES) for supporting the visit to Bures-sur-Yvette (IHES, France) in September–December 2003. The authors also acknowledge with gratitude the helpful comments and discussions of Professors I. V. Volovich and A. Yu. Khrennikov. The authors also express their gratitude to the referee for some useful observations.

PY - 2004/10

Y1 - 2004/10

N2 - We consider a nearest-neighbor p-adic λ-model with spin values ±1 on a Cayley tree of order k ≥ 1. We prove for the model there is no phase transition and as well as being unique, the p-adic Gibbs measure is bounded if and only if p ≥ 3. If p = 2, then we find a condition which guarantees the nonexistence of a phase transition. Besides, the results are applied to the p-adic Ising model and we show that for the model there is a unique p-adic Gibbs measure.

AB - We consider a nearest-neighbor p-adic λ-model with spin values ±1 on a Cayley tree of order k ≥ 1. We prove for the model there is no phase transition and as well as being unique, the p-adic Gibbs measure is bounded if and only if p ≥ 3. If p = 2, then we find a condition which guarantees the nonexistence of a phase transition. Besides, the results are applied to the p-adic Ising model and we show that for the model there is a unique p-adic Gibbs measure.

KW - Cayley tree

KW - Gibbs measure

KW - Ising model

KW - p-adic field

KW - λ-model

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JO - Letters in Mathematical Physics

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On the uniqueness of Gibbs measures for p-adic nonhomogeneous λ-model on the Cayley tree

Abstract

Keywords

ASJC Scopus subject areas

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