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

#### D. Gandolfo, U.A. Rozikov, J. Ruiz

2012, v.18, Issue 4, 701-720

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 exist 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 $p\ne 3$.

Keywords: Cayley trees,hard core interaction,Gibbs measures,translationinvariant measures,periodic measures,splitting measures,$p$-adic numbers