Holliday junctions in the Blume–Capel model of DNA

We regard a DNA molecule as a configuration of the Blume–Capel model on paths in a Cayley tree. We study translation-invariant Gibbs measures (TIGMs) of the model on the Cayley tree of order two and show that there is a critical temperature $T_\mathrm{c}$ such that there exists a unique TIGM if the temperature $T>T_\mathrm{c}$, there are two TIGMs if $T=T_\mathrm{c}$, and there are three TIGMs if $T<T_\mathrm{c}$. Each such measure describes a phase of the set of DNA molecules. We use these measures to study probability distributions of Holliday junctions in DNA molecules.