P-adic Gibbs Measures Research Articles

On set of p-adic Gibbs measures for the countable state 1D SOS model

On set of p-adic Gibbs measures for the countable state 1D SOS model

Phase Transition of Mixed Type p-Adic λ-Ising Model on Cayley Tree

In the present paper, we consider an interaction of the nearest-neighbors and next nearest-neighbors for the mixed type p-adic λ-Ising model with spin values {−1, +1} on the Cayley tree of order two.We obtained the uniqueness and existence of the p-adic quasi Gibbs measures for the model. Thereafter, as a main result, we proved the occurrence of phase transition for the p-adic λ-Ising model on the Cayley tree of order two. To establish the results, we employed some properties of p-adic numbers. Therefore, our results are not valid in the real case.

Periodic p-adic Gibbs Measures of q-State Potts Model on Cayley Trees I: The Chaos Implies the Vastness of the Set of p-Adic Gibbs Measures

We study the set of p-adic Gibbs measures of the q-state Potts model on the Cayley tree of order three. We prove the vastness of the set of the periodic p-adic Gibbs measures for such model by showing the chaotic behavior of the corresponding Potts–Bethe mapping over $$\mathbb {Q}_p$$ for the prime numbers $$p\equiv 1 (\mathrm {mod} 3)$$ . In fact, for $$0< |\theta -1|_p< |q|_p^2 < 1$$ where $$\theta =\exp _p(J)$$ and J is a coupling constant, there exists a subsystem that is isometrically conjugate to the full shift on three symbols. Meanwhile, for $$0< |q|_p^2 \le |\theta -1|_p< |q|_p < 1$$ , there exists a subsystem that is isometrically conjugate to a subshift of finite type on r symbols where $$r \ge 4$$ . However, these subshifts on r symbols are all topologically conjugate to the full shift on three symbols. The p-adic Gibbs measures of the same model for the prime numbers $$p=2,3$$ and the corresponding Potts–Bethe mapping are also discussed. On the other hand, for $$0< |\theta -1|_p< |q|_p < 1,$$ we remark that the Potts–Bethe mapping is not chaotic when $$p=3$$ and $$p\equiv 2 (\mathrm {mod} 3)$$ and we could not conclude the vastness of the set of the periodic p-adic Gibbs measures. In a forthcoming paper with the same title, we will treat the case $$0< |q|_p \le |\theta -1|_p < 1$$ for all prime numbers p.

P-Adic solid-on-solid model on a Cayley tree

We consider a p-adic solid-on-solid (SOS) model with a nearest-neighbor coupling, m+1 spins, and a coupling constant J ∈ Q p on a Cayley tree. We find conditions under which a phase transition does not occur in the model. We show that if p | m + 1 for some J, then a phase transition occurs. Moreover, we formulate a criterion for the boundedness of p-adic Gibbs measures for the (m+1)-state SOS model.

On chaotic behaviour of the p-adic generalized Ising mapping and its application

In the present paper, by conducting research on the dynamics of the p-adic generalized Ising mapping corresponding to renormalization group associated with the p-adic Ising-Vannemenus model on a Cayley tree, we have determined the existence of the fixed points of a given function. Simultaneously, the attractors of the dynamical system have been found. We have come to a conclusion that the considered mapping is topologically conjugate to the symbolic shift which implies its chaoticity and as an application, we have established the existence of periodic p-adic Gibbs measures for the p-adic Ising-Vannemenus model.

On the solvability of general cubic equations over ℤp*

The p-adic models of statistical mechanics require an investigation of the roots of polynomial equations over p-adic fields in order to construct p-adic Gibbs measures. The most frequently asked question is whether a root of a polynomial equation belongs to some given domains. In this paper, we study the solvability of general cubic equations over Z(p)* where prime p > 3. Our investigation enables us to describe all translation invariant p-adic Gibbs measures on a Cayley tree of order three.

Constraction of p-Adic Gibbs Measure for p-Adic λ-Ising Model

In this work we establish the existence of the p-adic Gibbs measure for the p-adic λ-Ising model on the Cayley tree of order two. In the previous studies, p-adic Ising model and p-adic λ-model were distinctly studied in many papers with the various properties. In this work, it is the first we combined both p-adic Ising model and p-adic λ-model on the Cayley tree of order two. In this research we only establish the model and we proved the existence of the p-adic Gibbs measures in the p-adic case. Here, we use the methods of p-adic analysis, and then, our results do not work in the real case.

A p-adic hard-core model with three states on a Cayley tree

We examine the p-adic hard-core model with three states on a Cayley tree. Translationinvariant and periodic p-adic Gibbs measures are studied for the hard-core model for k = 2. We prove that every p-adic Gibbs measure is bounded for p ≠ 2. We show in particular that there is no strong phased transition for a hard-core model on a Cayley tree of order k.

On periodic Gibbs measures of p-adic Potts model on a Cayley tree

In the present paper, we study the existence of periodic p-adic quasi Gibbs measures of p-adic Potts model over the Cayley tree of order two. We first prove that the renormalized dynamical system associated with the model is conjugate to the symbolic shift. As a consequence of this result we obtain the existence of countably many periodic p-adic Gibbs measures for the model.

Non Uniqueness of p-adic Gibbs Distribution for the Ising Model on the Lattice Zd

Real Gibbs measures arise in many problems of probability theory and statistical mechanics. This measure, related to the Boltzmann distribution, generalizes the notion of canonical ensemble. In addition, Gibbs measure is unique measure that maximizes the entropy of the expected energy. But non-archimedean (p-adic) analogue of Gibbs measures have been little studied. It is known that in the case of real numbers concepts of Gibbs measure and Markov random field are identical. But in the p-adic case, the class of p-adic Markov random fields is wider than the class of p-adic Gibbs measures [1]. One of the main problems of physics is to study the set of all p-adic Gibbs measures (see e.g. [1, 2]). Let us present some main definitions from the theory of p-adic numbers (see [3–5]). Let p be a prime number. Every rational number x = 0 can be represented in the form x = p n m , where r, n ∈ Z, m is a positive number, (n,m) = 1, where m and n are not divisible by p. A p-adic norm of rational number x = p n m is defined as follows

On p-adic Ising–Vannimenus model on an arbitrary order Cayley tree

In this paper, we continue an investigation of the p-adic Ising–Vannimenus model on the Cayley tree of an arbitrary order k(k ⩾ 2). We prove the existence of p-adic quasi Gibbs measures by analyzing fixed points of multi-dimensional p-adic system of equations. We are also able to show the uniqueness of translation-invariant p-adic Gibbs measure. Finally, it is established the existence of the phase transition for the Ising–Vannimenus model depending on the order k of the Cayley tree and the prime p. Note that the methods used in the paper are not valid in the real setting, since all of them are based on p-adic analysis and p-adic probability measures.

On P-Adic λ-Model on the Cayley Tree II: Phase Transitions

In the present paper, we consider the p-adic λ-model on the Cayley tree of order two. It is considered generalized p-adic quasi Gibbs measure depending on a parameter ρ ∈ ℚp, for the λ-model. In the p-adic setting there are several kinds of phase transitions such as storing phase transition, phase transition in terms of the generalized p-adic quasi Gibbs measures. In the paper, we consider two regimes with respect to the parameter ρ. The existence of generalized p-adic Gibbs measures in both regimes is proved. We prove the existence of the phase transition for the p-adic λ model on the Cayley tree of order two in the first regime. It turns out that in the second regime, we are able to establish the strong phase transition for a class of λ-models on the same tree. To prove the main results, we employ the methods of p-adic analysis, and therefore, our results are not valid in the real setting.

Quadratic equations over p-adic fields and their applications in statistical mechanics

The p-adic models of statistical mechanics require the investigation of the roots of polynomial equations over p-adic fields in order to construct p-adic Gibbs measures. The most frequently asked question is that whether a root of a polynomial equation belongs to the classic domains not. This question was open even for a quadratic equation. In this paper, by using the Newton polygon, we provide solvability criteria for quadratic equations over the domains mentioned above for all odd primes p. We also study the number of roots of quadratic equations over all domains given above. This study allows us to present a local description of roots of quadratic equations over p-adic fields whenever p > 2.

On non-Archimedean recurrence equations and their applications

In the present paper we study stability of recurrence equations (which in particular case contain dynamics of rational functions) generated by contractive functions defined on an arbitrary non-Archimedean algebra. Moreover, multirecurrence equations are considered. We also investigate reverse recurrence equations which have application in the study of p-adic Gibbs measures. Note that our results also provide the existence of unique solutions of nonlinear functional equations. We should stress that the non-Archimedeanity of the algebra is essentially used, therefore, the methods applied in the present paper are not valid in the Archimedean setting.

On a generalized p-adic Gibbs measure for Ising model on trees

In this paper we consider a p-adic Ising model on an arbitraty tree. We show the uniqueness and boundedness of the p-adic Gibbs measure for the model. Moreover, we consider translational invariant and periodic generalized p-adic Gibbs measures for the model on the Cayley tree of order two.

On the strong phase transition for the one-dimensional countable state p-adic Potts model

In the present paper we consider the countable state p-adic Potts model on . A main aim is to establish the existence of the strong phase transition for the model. In our study, we essentially use one dimensionality of the model. To prove the existence of the phase transition, we reduce the problem to the investigation of an infinite-dimensional nonlinear equation. We find a condition on weights to show that the derived equation has two solutions. We show that measures corresponding to first and second solutions are a p-adic Gibbs and generalized p-adic Gibbs measures, respectively. Moreover, it is proved that the p-adic Gibbs measure is bounded, and the generalized one is not bounded. This implies the existence of the strong phase transition. Note that it turns out that the obtained 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. Note that, in the usual real case, if one considers a one-dimensional translation-invariant model with nearest neighbor interaction, then such a model does not exhibit a phase transition. Nevertheless, we should stress that our model exhibits a unique p-adic Gibbs measure.

P-adic Gibbs measures for the hard core model with three states on the Cayley tree

We study p-adic hard core models with three states on the Cayley tree. It is known that there are four types of such models. We find conditions that must be imposed on the order k of the Cayley tree and on the prime p for a translation-invariant p-adic Gibbs measure to exist.

The p-adic Potts model on the Cayley tree of order three

We study a phase transition problem for the q-state p-adic Potts model on the Cayley tree of order three. We find certain conditions for the existence of p-adic Gibbs measures and then establish the existence of a phase transition.

On p-adic Gibbs measures for Ising model with four competing interactions

In this paper we consider a p-adic Ising model with four competing interactions (nearest neighbor, second neighbors and triples of neighbors) on the Cayley tree of order two. We show the uniqueness of the p-adic Gibbs measure for the model. Moreover, we consider p-adic quasi Gibbs measure for the model.

Phase transitions for p-adic Potts model on the Cayley tree of order three

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 ρ∈Qp. Such a measure is called generalized p-adic quasi Gibbs measure. When ρ equals the p-adic exponent, then it coincides with the p-adic Gibbs measure. When ρ = p, then it coincides with the 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 = expp(J) for some J∈Qp, 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 |ρ|p,|q|p ≤ p−2 we prove the existence of a quasi phase transition. It turns out that if and , then one finds the existence of the strong phase transition.