$p$-Adic Gibbs measures and Markov random fields on countable graphs

The notions of the Gibbs measure and of the Markov random field are known to coincide in the real case. But in the $p$-adic case, the class of $p$-adic Markov random fields is broader than that of $p$-adic Gibbs measures. We construct $p$-adic Markov random fields (on finite graphs) that are not $p$-adic Gibbs measures. We define a $p$-adic Markov random field on countable graphs and show that the set of such fields is a nonempty closed subspace in the set of all $p$-adic probability measures.