The p-adic Gibbs measure for the four-state stick-Hard-Core model on a Cayley tree

This study explores the existence and uniqueness of p-adic Gibbs measures for the p-adic G-HC model on the Cayley tree with order k ≥ 1, where G represents the stick graph. We first identify a necessary condition that restricts the field (Qₚ) in which solutions can exist based on the Cayley tree order. Leveraging this condition, we establish a sufficient condition guaranteeing the existence of a p-adic Gibbs measure. Finally, we prove the uniqueness of the translation-invariant p-adic Gibbs measure for the considered model when these conditions are satisfied.