Phase transitions for P -adic Potts model on the Cayley tree of order three
Abstract
In the present paper, we study a phase transition problem for the
q
-state
p
-adic Potts model over the Cayley tree of order three. We consider a more general notion of
p
-adic Gibbs measure which depends on parameter $\rho\in\bq_p$. Such a measure is called {\it generalized
p
-adic quasi Gibbs measure}. When
ρ
equals to
p
-adic exponent, then it coincides with the
p
-adic Gibbs measure. When
ρ=p
, then it coincides with
p
-adic quasi Gibbs measure. Therefore, we investigate two regimes with respect to the value of
|ρ
|
p
. Namely, in the first regime, one takes
ρ=
exp
p
(J)
for some $J\in\bq_p$, in the second one
|ρ
|
p
<1
. In each regime, we first find conditions for the existence of generalized
p
-adic quasi Gibbs measures. Furthermore, in the first regime, we establish the existence of the phase transition under some conditions. In the second regime, when $|\r|_p,|q|_p\leq p^{-2}$ we prove the existence of a quasi phase transition. It turns out that if $|\r|_p<|q-1|_p^2<1$ and $\sqrt{-3}\in\bq_p$, then one finds the existence of the strong phase transition.