Saburov, Mansoor; Khameini Ahmad, Mohd Ali On descriptions of all translation invariant \(p\)-adic Gibbs measures for the Potts model on the Cayley tree of order three. (English) Zbl 1456.82181 Math. Phys. Anal. Geom. 18, No. 1, Article ID 26, 33 p. (2015). Summary: 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 \(\mathbb Z_p\setminus\mathbb Z_p^\ast\). In own its turn, we also provide a solvability criterion of a general cubic equation over \(\mathbb Z_p\setminus\mathbb Z_p^\ast\) for \(p>3\). Cited in 1 ReviewCited in 8 Documents MSC: 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics 82B26 Phase transitions (general) in equilibrium statistical mechanics 05C05 Trees 11D25 Cubic and quartic Diophantine equations 11D88 \(p\)-adic and power series fields Keywords:\(p\)-adic number; \(p\)-adic Potts model; \(p\)-adic Gibbs measure PDFBibTeX XMLCite \textit{M. Saburov} and \textit{M. A. Khameini Ahmad}, Math. Phys. Anal. Geom. 18, No. 1, Article ID 26, 33 p. 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