Lattice models with interactions on Caylay tree

We consider an Ising competitive model defined over a triangular Husimi tree where loops, responsible for an explicit frustration, are even allowed. We first analyze the phase diagram of the model with fixed couplings in which a “gas of noninteracting dimmers (or spin liquid) — ferro or antiferromagnetic ordered state” zero temperature transition is recognized in the frustrated regions. Then we introduce the disorder for studying the spin glass version of the model: the triangular ±J model. We find out that, for any finite value of the averaged couplings, the model exhibits always a finite temperature phase transition even in the frustrated regions, where the transition turns out to be a glassy transition. On the other hand, In this investigation we studied one-dimensional countable state p-adic Potts model. We prove the existence of generalized p-adic Gibbs measures for the given model. It is also shown that under the condition there may occur a phase transition