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

Abstract

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, $z={\mathrm{e}}^{-h/T}$ and $t={\mathrm{e}}^{-J/T}$ 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 $n\to \infty$. 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 finitely many 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.

32 pages, 15 figures. Comments welcome! Updated version fixes some typos, clarifies some proofs, and includes new Remarks 1 and 7