In this paper we consider a nearest-neighbor p-adic hard core (HC) model,
with fugacity λ, 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
2k−1. Moreover if p divides 2k−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≥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 λ 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
pî€ =3.Comment: 17 page