On the Existence of Phase Transition for the 1D $p$-Adic Countable State Potts Model

In the present paper we establish the existence of a phase transition for one-dimensional the countable state $p$-adic Potts model. To establish such results, we investigate an infinite dimensional nonlinear $p$-adic operator associated with the model. It turns out that the finding condition does not depend on values of the prime p, and therefore, an analogous fact is not true when the number of spins is finite.