The description of generalized translation-invariant \(p\)-adic Gibbs measures for the Potts model on the Cayley tree of order three

M. Alp^a^,*, Chin Hee Pah^b^,**, and M. K. Saburov^a^,***

^aCollege of Engineering and Technology, American University of the Middle East, Kuwait City, Kuwait

^bDepartment of Computational and Theoretical Sciences, Faculty of Science, International Islamic University Malaysia, Pahang, Kuantan, Malaysia

email: *murat.alp@aum.edu.kw
email: **pahchinhee@iium.edu.my
email: ***mansur.saburov@aum.edu.kw

Received 18 September, 2022

Abstract— We describe generalized translation-invariant \(p\)-adic Gibbs measures for the Potts model on the Cayley tree of order three, confirming and extending the related results in the literature. We introduce a cube root function over the \(p\)-adic field \(\mathbb{Q}_p\), which enables us to explicitly prescribe all roots of the cubic monomial equation over \(\mathbb{Q}_p\).

Keywords: Gibbs measure, Potts model, Cayley tree, \(p\)-adic field

DOI: 10.1134/S0040577923030066