On $p$-adic Gibbs Measures for Hard Core Model on a Cayley Tree

TL;DR

This paper investigates the properties of $p$-adic Gibbs measures for the hard core model on Cayley trees, revealing unique divisibility conditions and existence criteria that distinguish the $p$-adic case from the classical real case.

Contribution

It provides a comprehensive analysis of $p$-adic Gibbs measures for the HC model, including existence, uniqueness, and periodicity conditions based on divisibility properties of $p$ and $k$, and highlights differences from real models.

Findings

Splitting Gibbs measures exist only if $p$ divides $2^k-1$.

Unique translational invariant measure exists under certain divisibility conditions.

For $k=2$, $p$-adic splitting Gibbs measures exist only if $p=3$, with two non-translational measures when $ ext{radius} o 1/27$.

Abstract

In this paper we consider a nearest-neighbor $p$-adic hard core (HC) model, with fugacity $\lambda$, on a homogeneous Cayley tree of order $k$ (with $k + 1$ neighbors). We focus on $p$-adic Gibbs measures for the HC model, in particular on $p$-adic "splitting" Gibbs measures generating a $p$-adic Markov chain along each path on the tree. We show that the $p$-adic HC model is completely different from real HC model: For a fixed $k$ we prove that the $p$-adic HC model may have a splitting Gibbs measure only if $p$ divides $2^k-1$. Moreover if $p$ divides $2^k-1$ but does not divide $k+2$ then there exists unique translational invariant $p$-adic Gibbs measure. We also study $p$-adic periodic splitting Gibbs measures and show that the above model admits only translational invariant and periodic with period two (chess-board) Gibbs measures. For $p\geq 7$ (resp. $p=2,3,5$) we give necessary and sufficient (resp. necessary) conditions for the existence of a periodic $p$-adic measure. For k=2 a $p$-adic splitting Gibbs measures exists if and only if p=3, in this case we show that if $\lambda$ belongs to a $p$-adic ball of radius 1/27 then there are precisely two periodic (non translational invariant) $p$-adic Gibbs measures. We prove that a $p$-adic Gibbs measure is bounded if and only if $p\ne 3$.