On \(p\)-adic Gibbs measures for hard core model on a Cayley tree.

*(English)*Zbl 1281.82006Summary: 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 |