A dynamical approach to studying the Lee-Yang zeros for the Potts Model on the Cayley Tree

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, $z={\mathrm{e}}^{-h\mathrm{/}T}$ and $t={\mathrm{e}}^{-J\mathrm{/}T}$ 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 $t\in (0,\mathrm{\infty })$ and $z\in (0,\mathrm{\infty })$. For any fixed $t_0\in (0,\mathrm{\infty })$ and fixed $n\in \mathbb{N}$ we consider the complex zeros of $Z_n(z,t_0)$ and how they accumulate on the ray $(0,\mathrm{\infty })$ of physical values for $z$ as $n\to \mathrm{\infty }$. In the ferromagnetic case ($J>0$ or equivalently $t\in (0,1)$) these Lee-Yang zeros accumulate to at most one point on $(0,\mathrm{\infty })$ which we describe using explicit formulae. In the antiferromagnetic case $(J<0$ or equivalently $t\in (1,\mathrm{\infty })$) these Lee-Yang zeros accumulate to finitely many points of $(0,\mathrm{\infty })$, 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.