Renormalization method in $p$-adic $\lambda$-model on the Cayley tree

Abstract

In this present paper, it is proposed the renormalization techniques in the investigation of phase transition phenomena in $p$-adic statistical mechanics. We mainly study $p$-adic $ł$-model on the Cayley tree of order two. We consider generalized $p$-adic quasi Gibbs measures depending on parameter $\r\in\bq_p$, for the $ł$-model. Such measures are constructed by means of certain recurrence equations. These equations define a dynamical system. We study two regimes with respect to parameters. In the first regime we establish that the dynamical system has one attractive and two repelling fixed points, which predicts the existence of a phase transition. In the second regime the system has two attractive and one neutral fixed points, which predicts the existence of a quasi phase transition. A main point of this paper is to verify (i.e. rigorously prove) and confirm that the indicated predictions (via dynamical systems point of view) are indeed true. To establish the main result, we employ the methods of $p$-adic analysis, and therefore, our results are not valid in the real setting.

18 pages

References in corpus (12)

• The quantum structure of spacetime at the Planck scale and quantum fields
• p-Adic Mathematical Physics
• Application of p-adic analysis to models of spontaneous breaking of the replica symmetry
• Cayley Trees and Bethe Lattices, a concise analysis for mathematicians and physicists
• On $p$-adic Gibbs measures of countable state Potts model on the Cayley tree
• Ultrametric pseudodifferential operators and wavelets for the case of non homogeneous measure
• Generalized probabilities taking values in non-Archimedean fields and topological groups
• On Uniqueness of Gibbs Measures for $P$-Adic Nonhomogeneous $ł$-Model on the Cayley Tree
• Linear Fractional p-Adic and Adelic Dynamical Systems
• On non-Archimedean recurrence equations and their applications
• Renormalization Group: Applications in Statistical Physics
• On the chaotic behavior of a generalized logistic $p$-adic dynamical system

Cited by in corpus (2)

• On chaotic behavior of the $P$-adic generalized Ising mapping and its application
• The Dynamics of The Potts-Bethe Mapping over ${\mathbb{Q}}_p:$ The Case $p\equiv 2(mod3)$