Let \(Z_n(z,t)\) denote the partition function of the q-state Potts Model on the rooted binary Cayley tree of depth n. Here, \(z = \textrm{e}^{-h/T}\) and \(t = \textrm{e}^{-J/T}\) with h denoting an externally applied magnetic field, T the temperature, and J a coupling constant. One can interpret z as a “magnetic field-like” variable and t as a “temperature-like” variable. Physical values \(h \in \mathbb {R}, T > 0\) , and \(J \in \mathbb {R}\) correspond to \(t \in (0,\infty )\) and \(z \in (0,\infty )\) . For any fixed \(t_0 \in (0,\infty )\) and fixed \(n \in \mathbb {N}\) we consider the complex zeros of \(Z_n(z,t_0)\) and how they accumulate on the ray \((0,\infty )\) of physical values for z as \(n \rightarrow \infty \) . In the ferromagnetic case ( \(J >0\) or equivalently \(t \in (0,1)\) ) these Lee-Yang zeros accumulate to at most one point on \((0,\infty )\) which we describe using explicit formulae. In the antiferromagnetic case \((J < 0\) or equivalently \(t \in (1,\infty )\) ) these Lee-Yang zeros accumulate to at most two points of \((0,\infty )\) , which we again describe with explicit formulae. The same results hold for the unrooted Cayley tree of branching number two. These results are proved by adapting a renormalization procedure that was previously used in the case of the Ising model on the Cayley Tree by Müller-Hartmann and Zittartz (1974 and 1977), Barata and Marchetti (1997), and Barata and Goldbaum (2001). We then use methods from complex dynamics and, more specifically, the active/passive dichotomy for iteration of a marked point, along with detailed analysis of the renormalization mappings, to prove the main results.
