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On $$p$$-adic Gibbs measures for hard core model on a Cayley tree. (English) Zbl 1281.82006
Summary: We consider a nearest-neighbour $$p$$-adic hard core (HC) model with fugacity $$\lambda$$, on a homogenous Cayley tree of order $$k$$ (with $$k+1$$ neighbours). We focus on $$p$$-adic gubbs 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 the 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 a 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 measure 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 $$p \neq 3$$.

##### MSC:
 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics 82B26 Phase transitions (general) in equilibrium statistical mechanics 12J12 Formally $$p$$-adic fields 46S10 Functional analysis over fields other than $$\mathbb{R}$$ or $$\mathbb{C}$$ or the quaternions; non-Archimedean functional analysis 60K35 Interacting random processes; statistical mechanics type models; percolation theory 05C05 Trees