Translation-invariant $p$-adic quasi-Gibbs measures for the Ising–Vannimenus model on a Cayley tree

We consider the $p$-adic Ising–Vannimenus model on the Cayley tree of order $k=2$. This model contains nearest-neighbor and next-nearest-neighbor interactions. We investigate the model using a new approach based on measure theory (in the $p$-adic sense) and describe all translation-invariant $p$-adic quasi-Gibbs measures associated with the model. As a consequence, we can prove that a phase transition exists in the model. Here, "phase transition" means that there exist at least two nontrivial $p$-adic quasi-Gibbs measures such that one is bounded and the other is unbounded. The methods used are inapplicable in the real case.