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Abstract <p>We study the Hard-Core model with a countable set <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb{Z}\)</EquationSource> <!--LobJMat2560742Khakimov-m1--> </InlineEquation> of spin values and a countable set of parameters <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\lambda_{i}&gt;0\)</EquationSource> <!--LobJMat2560742Khakimov-m2--> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(i\in\mathbb{Z}\)</EquationSource> <!--LobJMat2560742Khakimov-m3--> </InlineEquation> on a Cayley tree. In the case the normal subgroup of index four, we show the uniqueness of weakly periodic Gibbs measures under certain conditions on the parameters for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\lambda&lt;+\infty\)</EquationSource> <!--LobJMat2560742Khakimov-m4--> </InlineEquation>, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\lambda=\sum_{i}\lambda_{i}\)</EquationSource> <!--LobJMat2560742Khakimov-m5--> </InlineEquation>. Moreover, under some conditions we prove the existence of the weakly periodic (non periodic) Gibbs measures which are different from the known weakly periodic measures.</p>

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Uniqueness and Nonuniqueness Conditions for Weakly Periodic Gibbs Measures for the Hard-core Model with a Countable Set of Spin Values

  • 期刊论文

  • 发表日期：

    2025年9月1日

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  • R. M. Khakimov,
  • A. A. Bozorkulov

摘要

Abstract

We study the Hard-Core model with a countable set \(\mathbb{Z}\) of spin values and a countable set of parameters \(\lambda_{i}>0\) , \(i\in\mathbb{Z}\) on a Cayley tree. In the case the normal subgroup of index four, we show the uniqueness of weakly periodic Gibbs measures under certain conditions on the parameters for \(\lambda<+\infty\) , where \(\lambda=\sum_{i}\lambda_{i}\) . Moreover, under some conditions we prove the existence of the weakly periodic (non periodic) Gibbs measures which are different from the known weakly periodic measures.