A dynamical approach to studying the Lee-Yang zeros for the Potts Model on the Cayley Tree
Diyath Pannipitiya, Roland Roeder
TL;DR
This work analyzes Lee-Yang zeros for the $q$-state Potts model on binary Cayley trees through a dynamical-systems lens. By constructing renormalization maps $R_{z,t,q}$ and $\hat R_{z,t,q}$ and employing the active/passive dichotomy, the authors precisely characterize the accumulation loci of zeros on the physical $z$-ray as the depth tends to infinity, yielding explicit thresholds $t_1(q)$, $t_2(q)$, and $t_3(q)$ and accumulation points $z_c(t,q)$ and $z_c^{\pm}(t,q)$ for ferromagnetic and antiferromagnetic regimes (with special cases for $q=2$). The analysis extends Ising results to Potts models (including unrooted trees) and provides a rigorous dynamical framework (Theorem D) linking zeros to iterates of renormalization maps, complemented by numerical plots and detailed lemmas. The methodology advances the mathematical understanding of phase-transition indicators in hierarchical lattices and offers a robust approach for studying Lee-Yang zeros via complex dynamics. Overall, the paper delivers explicit accumulation descriptions, rigorous proofs, and a unifying dynamical-system perspective for Potts-model zeros on Cayley trees.
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, $z = {\rm e}^{-h/T}$ and $t = {\rm e}^{-J/T}$ 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 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.