On free energies of the Ising model on the Cayley tree

TL;DR

This paper derives explicit formulas for the free energies and entropies of the Ising model on the Cayley tree under various boundary conditions, including translation-invariant, periodic, and weakly periodic states.

Contribution

It provides new explicit formulas for free energies of the Ising model on the Cayley tree for multiple boundary conditions, including recently discovered weakly periodic states.

Findings

Explicit free energy formulas for various boundary conditions.

Analysis of weakly periodic Gibbs states and their properties.

Density calculations for the 4-edge-coloring partition.

Abstract

We present, for the Ising model on the Cayley tree, some explicit formulae of the free energies (and entropies) according to boundary conditions (b.c.). They include translation-invariant, periodic, Dobrushin-like b.c., as well as those corresponding to (recently discovered) weakly periodic Gibbs states. The later are defined through a partition of the tree that induces a 4-edge-coloring. We compute the density of each color.