Zero-Freeness is All You Need: A Weitz-Type FPTAS for the Entire Lee-Yang Zero-Free Region
Shuai Shao, Ke Shi
TL;DR
The paper develops a Weitz-type FPTAS for the ferromagnetic Ising model across the full Lee–Yang zero-free region without assuming strong spatial mixing. It achieves this by replacing vertex marginal computations with edge-deletion ratios and evaluating truncated Taylor series of these ratios via the SAW-tree, together with zero-freeness to control errors. A key innovation is the local dependence of coefficients (LDC) and a combinatorial divisibility framework that yields edge-based SSM, which in turn implies SSM for the random-cluster model and enables broader applications to the Potts model, hypergraph independence polynomial, and Holant problems. The results provide both FPTAS algorithms and optimal mixing-time implications on lattices, significantly expanding the practical reach of Weitz-type approaches beyond regions where SSM is known to hold, and offering a unifying framework anchored in zero-freeness and LDC.
Abstract
We present a Weitz-type FPTAS for the ferromagnetic Ising model across the entire Lee--Yang zero-free region, without relying on the strong spatial mixing (SSM) property. Our algorithm is Weitz-type for two reasons. First, it expresses the partition function as a telescoping product of ratios, with the key being to approximate each ratio. Second, it uses Weitz's self-avoiding walk tree, and truncates it at logarithmic depth to give a good and efficient approximation. The key difference from the standard Weitz algorithm is that we approximate a carefully designed edge-deletion ratio instead of the marginal probability of a vertex being assigned a particular spin, ensuring our algorithm does not require SSM. Furthermore, by establishing local dependence of coefficients (LDC), we prove a novel form of SSM for these edge-deletion ratios, which, in turn, implies the standard SSM for the random cluster model. This is the first SSM result for the random cluster model on general graphs, beyond lattices. Our proof of LDC is based on a new divisibility relation, and we show such relations hold quite universally. This leads to a broadly applicable framework for proving LDC across a variety of models, including the Potts model, the hypergraph independence polynomial, and Holant problems. Combined with existing zero-freeness results for these models, we derive new SSM results for them.