| Summary: | In the present paper we shall consider countable state p -adic Potts model on Z . A main aim is to establish the existence of the phase transition for the model. In our study, we essentially use one dimensionality of the model. The existence of the phase transition is reduced to the investigation of an infinitedimensional nonlinear equation. We find a condition on weights to show that the
derived equation has two solutions, which yields the existence of the phase transition. We prove that measures corresponding to first and second solutions are a p -adic Gibbs and generalized p -adic Gibbs measures, respectively. Note that 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
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