zbMATH — the first resource for mathematics

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 \(p \neq 3\).

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