$\mathbb {Z}$Existence of a Phase Transition for the Potts $p$-adic Model on the Set $\mathbb {Z}$

We consider the Potts model on the set $\mathbb {Z}$ in the field $Q_p$ of $p$-adic numbers. The range of the spin variables $\sigma (n)$, $n\in \mathbb Z$, in this model is $\Phi =\{\sigma _1,\sigma _2,\dots \dots ,\sigma _q\}\subset Q_p^{q-1}=\underbrace {Q_p\times Q_p\times \dots \times Q_p}_{q-1}$. We show that there are some values $q=q(p)$ for which phase transitions.