Abstract
The seminal Lee-Yang theorem states that for any graph the zeros of the partition function of the ferromagnetic Ising model lie on the unit circle in $\mathbb C$. In fact the union of the zeros of all graphs is dense on the unit circle. In this paper we study the location of the zeros for the class of graphs of bounded maximum degree $d\geq 3$, both in the ferromagnetic and the anti-ferromagnetic case. We determine the location exactly as a function of the inverse temperature and the degree $d$. An important step in our approach is to translate to the setting of complex dynamics and analyze a dynamical system that is naturally associated to the partition function.
Highlights
Introduction and main resultFor a graph G = (V, E), ξ, b ∈ C, the partition function of the Ising model ZG(ξ, b) is defined as ZG(ξ, b) := ξ|U| · b|δ(U)|, (1.1)U ⊆V where δ(U ) denotes the collection of edges with one endpoint in U and one endpoint in V \ U
We focus on the collection of graphs of bounded degree, and completely describe the location of the zeros for this class of graphs
We remark that part (ii) has recently been independently proved by Chio, He, Ji and Roeder [9]. They focus on the class of Cayley trees and obtain a precise description of the limiting behavior of the zeros of the partition function of the Ising model
Summary
We remark that part (ii) has recently been independently proved by Chio, He, Ji and Roeder [9] They focus on the class of Cayley trees and obtain a precise description of the limiting behavior of the zeros of the partition function of the Ising model. In our setting, the existence of a zero-free disk normal to the unit circle containing the point ξ = +1 was proved by Lieb and Ruelle [15]. The value of αb can again be explicitly expressed in terms of b, see Figure 1 for an illustration depicting αb Another recent related contribution to the Lee–Yang program is due to Liu, Sinclair and Srivastava [18], who showed that for ξ = 1 and d 2 there exists an open set B ⊂ C containing the interval
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