Description of periodic extreme gibbs measures of some lattice models on the Cayley tree

Ганиходжаев, Насир Набиевич; Rozikov, U. A.

doi:10.1007/bf02634202

Cited by 125 publications

(76 citation statements)

References 7 publications

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“…[22,17] (and subsequent papers [23,24], [1,2,7,19,20,21]), we consider a special class S ⊂ G of Gibbs measures. We call them splitting Gibbs measures, to emphasize the fact that, in addition to the aforementioned Markov property, they satisfy the following condition: given values σ(x), x ∈ V n , of an admissible configuration σ ∈ Ω over set V n , its values σ(y) at sites y ∈ W n+1 are conditionally independent.…”

Section: Construction Of Splitting Gibbs Measuresmentioning

confidence: 99%

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A Three State Hard-Core Model on a Cayley Tree

Martin¹,

Rozikov²,

Suhov³

2005

JNMP

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We consider a nearest-neighbor hard-core model, with three states , on a homogeneous Cayley tree of order k (with k + 1 neighbors). This model arises as a simple example of a loss network with nearest-neighbor exclusion. The state σ(x) at each node x of the Cayley tree can be 0, 1 and 2. We have Poisson flow of calls of rate λ at each site x, each call has an exponential duration of mean 1. If a call finds the node in state 1 or 2 it is lost. If it finds the node in state 0 then things depend on the state of the neighboring sites. If all neighbors are in state 0, the call is accepted and the state of the node becomes 1 or 2 with equal probability 1/2. If at least one neighbor is in state 1, and there is no neighbor in state 2 then the state of the node becomes 1. If at least one neighbor is in state 2 the call is lost. We focus on 'splitting' Gibbs measures for this model, which are reversible equilibrium distributions for the above process. We prove that in this model, ∀ λ > 0 and k ≥ 1, there exists a unique translationinvariant splitting Gibbs measure µ * . We also study periodic splitting Gibbs measures and show that the above model admits only translation -invariant and periodic with period two (chess-board) Gibbs measures. We discuss some open problems and state several related conjectures.

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Section: Construction Of Splitting Gibbs Measuresmentioning

confidence: 99%

“…Note that (see [7]) there exists a one-to-one correspondence between the set V of vertexes of the Cayley tree of order k ≥ 1 and the group G k of the free products of k + 1 cyclic groups of the second order with generators a 1 , a 2 , ..., a k+1 .…”

Section: Description Of Periodic Splitting Gibbs Measuresmentioning

confidence: 99%

A Three State Hard-Core Model on a Cayley Tree

Martin¹,

Rozikov²,

Suhov³

2005

JNMP

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“…A complete analysis of this set is often a difficult problem. Many papers have been devoted to these studies when the underlying lattice is a Cayley tree (see for example, [14,22,23,26,64,77]). …”

Section: Introductionmentioning

confidence: 99%

On Dynamical Systems and Phase Transitions for q + 1-state p-adic Potts Model on the Cayley Tree

Mukhamedov

2012

Math Phys Anal Geom

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In the present paper, we study a new kind of p-adic measures for q + 1-state Potts model, called p-adic quasi Gibbs measure. For such a model, we derive a recursive relations with respect to boundary conditions. Note that we consider two mode of interactions: ferromagnetic and antiferromagnetic. In both cases, we investigate a phase transition phenomena from the associated dynamical system point of view. Namely, using the derived recursive relations we define a fractional p-adic dynamical system. In ferromagnetic case, we establish that if q is divisible by p, then such a dynamical system has two repelling and one attractive fixed points. We find basin of attraction of the fixed point. This allows us to describe all solutions of the nonlinear recursive equations. Moreover, in that case there exists the strong phase transition. If q is not divisible by p, then the fixed points are neutral, and this yields that the existence of the quasi phase transition. In antiferromagnetic case, there are two attractive fixed points, and we find basins of attraction of both fixed points, and describe solutions of the nonlinear recursive equation. In this case, we prove the existence of a quasi phase transition.

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“…There are many works devoted to several kind of partitions of groups (lattices) (see e.g. [1][2][3][4][5]7]). …”

Section: Introductionmentioning

confidence: 99%

Characterization of the normal subgroups of finite index for the group representation of a Cayley tree

Haydarov¹

2016

Nanosystems: Phys. Chem. Math.

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Description of periodic extreme gibbs measures of some lattice models on the Cayley tree

Cited by 125 publications

References 7 publications

A Three State Hard-Core Model on a Cayley Tree

A Three State Hard-Core Model on a Cayley Tree

On Dynamical Systems and Phase Transitions for q + 1-state p-adic Potts Model on the Cayley Tree

Characterization of the normal subgroups of finite index for the group representation of a Cayley tree

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