Gibbs measures for $p$-adic Hard-Core model with a countable set of spin values

In this talk we consider generalized Gibbs measure (GGM) for $p$-adic Hard-Core (HC) model with a countable set of spin values on a Cayley tree. We analyze $p$-adic functional equation which provides the compatibility condition for the finite-dimensional generalized Gibbs distributions. We define $p$-adic Gibbs distributions as limit of the consistent family of finite-dimensional generalized Gibbs distributions and show that, for our $p$-adic HC model on a Cayley tree, such a Gibbs distribution does not exist. Under some conditions on parameters $p$ and $\lambda_i$ we find the number of translation-invariant and two-periodic GGMs for the $p$-adic HC model on the Cayley tree of order two.