A near-optimal zero-free disk for the Ising model
Viresh Patel, Guus Regts, Ayla Stam
TL;DR
The paper studies zeros of the Ising partition function on graphs with bounded degree and proves a near-optimal zero-free disk for the univariate partition function by reformulating via the even-set generating function and a block-structured polymer method that exploits block structure. The main result identifies a zero-free disk D(n_Δ) with n_Δ = (1 - 1/√(2(Δ-1)))^2/(Δ-1) (equivalently (1 - o_Δ(1))/(Δ-1)), and a girth-dependent extension. It also develops a general recursion for Z_even on U-subgraphs and bound techniques via walk generating functions, and extends the framework to block polynomials and multivariate settings, linking to algorithmic consequences such as an FPTAS inside the disk and hardness outside. These contributions illuminate the interplay between phase-transition diagnostics and computational tractability for Ising-like partition functions on bounded-degree graphs.
Abstract
The partition function of the Ising model of a graph $G=(V,E)$ is defined as $Z_{\text{Ising}}(G;b)=\sum_{σ:V\to \{0,1\}} b^{m(σ)}$, where $m(σ)$ denotes the number of edges $e=\{u,v\}$ such that $σ(u)=σ(v)$. We show that for any positive integer $Δ$ and any graph $G$ of maximum degree at most $Δ$, $Z_{\text{Ising}}(G;b)\neq 0$ for all $b\in \mathbb{C}$ satisfying $|\frac{b-1}{b+1}| \leq \frac{1-o_Δ(1)}{Δ-1}$ (where $o_Δ(1) \to 0$ as $Δ\to \infty$). This is optimal in the sense that $\tfrac{1-o_Δ(1)}{Δ-1}$ cannot be replaced by $\tfrac{c}{Δ-1}$ for any constant $c > 1$ subject to a complexity theoretic assumption. To prove our result we use a standard reformulation of the partition function of the Ising model as the generating function of even sets. We establish a zero-free disk for this generating function inspired by techniques from statistical physics on partition functions of a polymer models. Our approach is quite general and we discuss extensions of it to a certain types of polymer models.