Translation-invariant $p$-adic quasi Gibbs measures for the Potts model with an external field on the Cayley tree

The study is focused on investigation of $p$-adic Gibbs measures for the $q$-state Potts model with an external field and determination of the conditions for the existence of a phase transition. In this work, we derive a functional equation that satisfies the compatibility condition for $p$-adic quasi-Gibbs measures on a Cayley tree of order $k\ge$ 2. Furthermore, we prove that if $|q|_p$ = 1 there exists a unique $p$-adic Gibbs measure for this model. Additionally, for the Potts model on a binary tree, we identify three $p$-adic quasi-Gibbs measures under specific circumstances: one bounded and two unbounded, which implies a phase transition.