On Descriptions of All Translation Invariant p−adic Gibbs Measures for the Potts Model on The Cayley Tree of Order Three

Abstract

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 $\mathbb {Z}_{p}\setminus \mathbb {Z}_{p}^{*}$ . In own its turn, we also provide a solvability criterion of a general cubic equation over $\mathbb {Z}_{p}\setminus \mathbb {Z}_{p}^{*}$ for p > 3.