On Gibbs measures of the Ising model on (k, m) -ary trees

In mathematical statistical mechanics, Gibbs measures have been designed to represent equilibrium states and to model phase transition. In the literature, Gibbs measures associated with the Ising model on Cayley trees have been extensively studied. However, the Ising model has not been studied on (i,k)-ary trees before. In this paper, the phase transition problem is investigated for the Ising model on (i,k)-ary trees. Specifically, the model on Γ(1,k) and Γ(2,3), respectively, displays three distinct translation-invariant Gibbs measures in both the ferromagnetic and anti-ferromagnetic states, whereas the traditional Ising model lacks translation-invariant Gibbs measures in the anti-ferromagnetic state. Moreover, non-translation invariant Gibbs measures are constructed as well. Furthermore, for the considered model, the standard free energy does not exist, in contrast to its existence over regular trees.