On descriptions of all translation invariant p−adic gibbs measures for the potts model on the cayley tree of order three

Unlike the real number field, a set of p−adic Gibbs measures of p−adic lattice models of statistical mechanics has a complex structure in a sense that it is strongly tied up with a Diophantine problem over p−adic fields. Recently, all translation-invariant p−adic Gibbs measures of the p−adic Potts...

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Main Authors: Saburov, Mansoor, Ahmad, Mohd Ali Khameini
格式: Article
語言:English
出版: Springer 2015
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在線閱讀:http://irep.iium.edu.my/49321/1/saburov2015.pdf
http://irep.iium.edu.my/49321/
http://link.springer.com/article/10.1007/s11040-015-9194-5
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總結:Unlike the real number field, a set of p−adic Gibbs measures of p−adic lattice models of statistical mechanics has a complex structure in a sense that it is strongly tied up with a Diophantine problem over p−adic fields. Recently, all translation-invariant p−adic Gibbs measures of the p−adic Potts model on the Cayley tree of order two were described by means of roots of a certain quadratic equation over some domain of the p−adic field. In this paper, we consider the same problem on the Cayley tree of order three. In this case, we show that all translation-invariant p−adic Gibbs measures of the p−adic Potts model can be described in terms of roots of some cubic equation over Zp \ Z∗p. In own its turn, we also provide a solvability criterion of a general cubic equation over Zp \ Z∗p for p > 3.