R. M. Khakimov, M. T. Makhammadaliev, U. A. Rozikov
{"title":"Cayley树上具有可计数自旋值集的hc模型的Gibbs测度","authors":"R. M. Khakimov, M. T. Makhammadaliev, U. A. Rozikov","doi":"10.1007/s11040-023-09453-w","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we study the HC-model with a countable set <span>\\(\\mathbb Z\\)</span> of spin values on a Cayley tree of order <span>\\(k\\ge 2\\)</span>. This model is defined by a countable set of parameters (that is, the activity function <span>\\(\\lambda _i>0\\)</span>, <span>\\(i\\in \\mathbb Z\\)</span>). A functional equation is obtained that provides the consistency condition for finite-dimensional Gibbs distributions. Analyzing this equation, the following results are obtained:</p><ul>\n <li>\n <p>Let <span>\\(\\Lambda =\\sum _i\\lambda _i\\)</span>. For <span>\\(\\Lambda =+\\infty \\)</span> there is no translation-invariant Gibbs measure (TIGM) and no two-periodic Gibbs measure (TPGM);</p>\n </li>\n <li>\n <p>For <span>\\(\\Lambda <+\\infty \\)</span>, the uniqueness of TIGM is proved;</p>\n </li>\n <li>\n <p>Let <span>\\(\\Lambda _\\textrm{cr}(k)=\\frac{k^k}{(k-1)^{k+1}}\\)</span>. If <span>\\(0<\\Lambda \\le \\Lambda _\\textrm{cr}\\)</span>, then there is exactly one TPGM that is TIGM;</p>\n </li>\n <li>\n <p>For <span>\\(\\Lambda >\\Lambda _\\textrm{cr}\\)</span>, there are exactly three TPGMs, one of which is TIGM.</p>\n </li>\n </ul></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"26 2","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2023-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-023-09453-w.pdf","citationCount":"2","resultStr":"{\"title\":\"Gibbs Measures for HC-Model with a Cuountable Set of Spin Values on a Cayley Tree\",\"authors\":\"R. M. Khakimov, M. T. Makhammadaliev, U. A. Rozikov\",\"doi\":\"10.1007/s11040-023-09453-w\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper, we study the HC-model with a countable set <span>\\\\(\\\\mathbb Z\\\\)</span> of spin values on a Cayley tree of order <span>\\\\(k\\\\ge 2\\\\)</span>. This model is defined by a countable set of parameters (that is, the activity function <span>\\\\(\\\\lambda _i>0\\\\)</span>, <span>\\\\(i\\\\in \\\\mathbb Z\\\\)</span>). A functional equation is obtained that provides the consistency condition for finite-dimensional Gibbs distributions. Analyzing this equation, the following results are obtained:</p><ul>\\n <li>\\n <p>Let <span>\\\\(\\\\Lambda =\\\\sum _i\\\\lambda _i\\\\)</span>. For <span>\\\\(\\\\Lambda =+\\\\infty \\\\)</span> there is no translation-invariant Gibbs measure (TIGM) and no two-periodic Gibbs measure (TPGM);</p>\\n </li>\\n <li>\\n <p>For <span>\\\\(\\\\Lambda <+\\\\infty \\\\)</span>, the uniqueness of TIGM is proved;</p>\\n </li>\\n <li>\\n <p>Let <span>\\\\(\\\\Lambda _\\\\textrm{cr}(k)=\\\\frac{k^k}{(k-1)^{k+1}}\\\\)</span>. If <span>\\\\(0<\\\\Lambda \\\\le \\\\Lambda _\\\\textrm{cr}\\\\)</span>, then there is exactly one TPGM that is TIGM;</p>\\n </li>\\n <li>\\n <p>For <span>\\\\(\\\\Lambda >\\\\Lambda _\\\\textrm{cr}\\\\)</span>, there are exactly three TPGMs, one of which is TIGM.</p>\\n </li>\\n </ul></div>\",\"PeriodicalId\":694,\"journal\":{\"name\":\"Mathematical Physics, Analysis and Geometry\",\"volume\":\"26 2\",\"pages\":\"\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2023-03-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s11040-023-09453-w.pdf\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Physics, Analysis and Geometry\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s11040-023-09453-w\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Physics, Analysis and Geometry","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s11040-023-09453-w","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Gibbs Measures for HC-Model with a Cuountable Set of Spin Values on a Cayley Tree
In this paper, we study the HC-model with a countable set \(\mathbb Z\) of spin values on a Cayley tree of order \(k\ge 2\). This model is defined by a countable set of parameters (that is, the activity function \(\lambda _i>0\), \(i\in \mathbb Z\)). A functional equation is obtained that provides the consistency condition for finite-dimensional Gibbs distributions. Analyzing this equation, the following results are obtained:
Let \(\Lambda =\sum _i\lambda _i\). For \(\Lambda =+\infty \) there is no translation-invariant Gibbs measure (TIGM) and no two-periodic Gibbs measure (TPGM);
For \(\Lambda <+\infty \), the uniqueness of TIGM is proved;
Let \(\Lambda _\textrm{cr}(k)=\frac{k^k}{(k-1)^{k+1}}\). If \(0<\Lambda \le \Lambda _\textrm{cr}\), then there is exactly one TPGM that is TIGM;
For \(\Lambda >\Lambda _\textrm{cr}\), there are exactly three TPGMs, one of which is TIGM.
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