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

TL;DR

This paper develops renormalization techniques to analyze phase transitions in a $p$-adic $1$-model on the Cayley tree, revealing different fixed point behaviors corresponding to phase and quasi-phase transitions.

Contribution

It introduces a rigorous $p$-adic dynamical systems approach to study phase transitions in the $p$-adic $1$-model, with new fixed point analysis.

Findings

In one regime, one attractive and two repelling fixed points indicate phase transition.

In another regime, two attractive and one neutral fixed points indicate quasi-phase transition.

Results are specific to $p$-adic analysis and do not apply in real number settings.

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 $\l$-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 $\l$-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.