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 $k \geq 1$, where $G$ represents the stick graph. We first identify a necessary condition that restricts the field ($\mathbb Q_p$) 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.