On $P$-adic Ising-Vannimenus model on an arbitrary order Cayley tree

TL;DR

This paper investigates the $p$-adic Ising-Vannimenus model on Cayley trees of arbitrary order, proving the existence and uniqueness of certain $p$-adic Gibbs measures and establishing phase transitions based on tree order and prime $p$.

Contribution

It introduces new $p$-adic analytical methods to analyze Gibbs measures and phase transitions in the Ising-Vannimenus model on Cayley trees, extending prior real-valued approaches.

Findings

Existence of $p$-adic quasi Gibbs measures

Uniqueness of translation-invariant $p$-adic Gibbs measure

Phase transition depending on tree order $k$ and prime $p$

Abstract

In this paper, we continue an investigation of the $p$-adic Ising-Vannimenus model on the Cayley tree of an arbitrary order $k$ $(k\geq 2$). 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.