English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We derive the strong spatial mixing property for the general 2-spin systemfrom zero-free regions of its partition function. We view the partitionfunction of the 2-spin system as a multivariate function over three complexparameters $(\beta, \gamma, \lambda)$, and we allow the zero-free regions of$\beta, \gamma$ or $\lambda$ to be of arbitrary shapes. As long as thezero-free region contains a positive point and it is a complex neighborhood of$\lambda=0$ when fixing $\beta, \gamma \in \mathbb{C}$, or a complexneighborhood of $\beta\gamma=1$ when fixing $\beta, \lambda\in \mathbb{C}$ or$\gamma, \lambda\in \mathbb{C}$ respectively, we are able to show that thecorresponding 2-spin system exhibits strong spatial mixing on such a region.The underlying graphs of the 2-spin system are not necessarily of boundeddegree, while are required to include graphs with pinned vertices. We provethis result by establishing a Christoffel-Darboux type identity for the 2-spinsystem on trees and using certain tools from complex analysis.  To our best knowledge, our result is general enough to turn all currentlyknown zero-free regions of the partition function of the 2-spin system wherepinned vertices are allowed into the strong spatial mixing property. Moreover,we extend our result to obtain strong spatial mixing for the ferromagneticIsing model (even with non-uniform external fields) from the celebratedLee-Yang circle theorem.

From Zero-Freeness to Strong Spatial Mixing via a Christoffel-Darboux Type Identity