Extremality of Gibbs measures for the DNA-Ising molecule model on the Cayley tree

We examine a model of a DNA-Ising molecule on a Cayley tree of order $k\ge$ 2. For this model, we derive a system of functional equations, where each positive solution corresponds to a Gibbs measure. On the general order Cayley tree, we can solve the model exactly. Specifically, we can find the exact value of the critical temperature Tc for any $k\ge$ 2 so that, if $T\ge T_c$, there is a unique translation-invariant Gibbs measure (TIGM), and if $T < T_c$, there are three TIGMs. We determine the model’s typical configurations and stationary distributions for high enough and low enough temperatures. The primary attention is focused on the systematic study of the structure of the set of the Gibbs measures. In this paper, we present a non-trivial adaptation of famous techniques, such as the Martinelli–Sinclair–Weitz criterion for determining the extremality of TIGMs and the Kesten–Stigum criterion for determining the non-extremality of TIGMs. One of the important contributions of this paper is the resolution of the extremality versus non-extremality regions for one of the TIGMs on a Cayley tree of the general order. For the other TIGMs, the extremality and non-extremality regions are determined on Cayley trees of orders up to 5.