Chaotic behavior of renormalization group of the p-adic Ising model on ary trees
Abstract
We study periodic p -adic (quasi-)Gibbs measures for the Ising model on ary trees, with a particular focus on the irregular ( 2 , 3 ) -ary tree. Unlike the classical setting of regular Cayley trees, where translation-invariant and periodic Gibbs measures are relatively well understood, the irregular geometry leads to new phenomena. We prove that for every prime p > 3 there exists at least one translation-invariant p -adic Gibbs measure. Moreover, when p ≡ 1 ( mod 4 ) we show that the model admits an infinite family of periodic p -adic quasi-Gibbs measures whose periods range over all integers m ≥ 1 , yielding a genuine phase transition. Our approach is based on a renormalization-group (RG) analysis: we identify the associated p -adic RG map, prove that its Julia set is a Cantor-type set, and show that the induced dynamics on this set is conjugate to a full shift, hence chaotic in the sense of Devaney. These results reveal sharp contrasts between real and p -adic statistical mechanical models and enrich the emerging theory of p -adic probability on hierarchical structures.