📝 SummarXiv

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's spin, ensuring our algorithm does not require SSM. Furthermore, by establishing local dependence of coefficients (LDC), we indeed 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. We prove LDC using a new division relation, and remarkably, such relations hold quite universally. As a result, we establish LDC for a variety of models. Combined with existing zero-freeness results for these models, we derive new SSM results for them. Our work suggests that both Weitz-type FPTASes and SSM can be derived from zero-freeness, while zero-freeness alone suffices for Weitz-type FPTASes, SSM additionally requires LDC, a combinatorial property independent of zero-freeness.