A Dynamical Approach to Studying the Lee-Yang Zeros for the Potts Model on the Cayley Tree

Let \ (Zₙ (z, t) \) denote the partition function of the q -state Potts Model on the rooted binary Cayley tree of depth n. Here, \ (z = e^-h/T\) and \ (t = 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 R, T > 0\), and \ (J R\) correspond to \ (t (0, ) \) and \ (z (0, ) \). For any fixed \ (t₀ (0, ) \) and fixed \ (n N\) we consider the complex zeros of \ (Zₙ (z, t₀) \) and how they accumulate on the ray \ ( (0, ) \) of physical values for z as \ (n \). In the ferromagnetic case (\ (J >0\) or equivalently \ (t (0, 1) \) ) these Lee-Yang zeros accumulate to at most one point on \ ( (0, ) \) which we describe using explicit formulae. In the antiferromagnetic case \ ( (J < 0\) or equivalently \ (t (1, ) \) ) these Lee-Yang zeros accumulate to at most two points of \ ( (0, ) \), 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.