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On dynamical systems and phase transitions for \(q + 1\)-state \(p\)-adic Potts model on the Cayley tree. (English) Zbl 1280.46047
The author studies phase transition problems for the \(q + 1\)-state nearest-neighbor \(p\)-adic Potts model over the Cayley tree of order two, where \(p\) be a fixed prime number. He defines a new class of \(p\)-adic measures, called \(p\)-adic quasi Gibbs measure corresponding to the \(p\)-adic Potts model. He considers \(p\)-adic probability theory approaches to study \(q + 1\)-state nearest-neighbor \(p\)-adic Potts models on Cayley trees. For the model, he obtains a recursive relation with respect to boundary conditions. Using the derived recursive relations he defines one-dimensional fractional \(p\)-adic dynamical systems.
In both ferromagnetic and antiferromagnetic cases, he investigates phase transition phenomena from the associated dynamical system point of view. In the ferromagnetic case, under some conditions, the author shows that there are three translation-invariant \(p\)-adic quasi Gibbs measures. He proves that if \(q\) is divisible by \(p\), then the dynamical system has two repelling and one attractive fixed points. He finds the basin of attraction of the attractive fixed point. Also, he proves that if \(q\) is not divisible by \(p\), then the fixed points are neutral, and this yields that the existence of the quasi phase transition. In the antiferromagnetic case, he obtains two attractive fixed points, and he finds the basins of attraction of both attractive fixed points. In this case, he proves the existence of a quasi phase transition. He shows that the transition depends on the number of spins \(q\). He studies boundedness and unboundedness of \(p\)-adic quasi Gibbs measures corresponding to the fixed points of the dynamical system. Therefore, he determines the existence of phase transitions, strong phase transitions or a quasi phase transitions.

MSC:
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
82B26 Phase transitions (general) in equilibrium statistical mechanics
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
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