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

Loop percolation, also known as the dense O(1) loop model, is a variant ofcritical bond percolation in the square lattice Z^2 whose graph structureconsists of a disjoint union of cycles. We study its connectivity pattern,which is a random noncrossing matching associated with a loop percolationconfiguration. These connectivity patterns exhibit a striking rationalityproperty whereby probabilities of naturally-occurring events are dyadicrational numbers or rational functions of a size parameter n, but the reasonsfor this are not completely understood. We prove the rationality phenomenon ina few cases and prove an explicit formula expressing the probabilities in the"cylindrical geometry" as coefficients in certain multivariate polynomials.This reduces the rationality problem in the general case to that of proving afamily of conjectural constant term identities generalizing an identity due toDi Francesco and Zinn-Justin. Our results make use of, and extend, algebraictechniques related to the quantum Knizhnik-Zamolodchikov equation.

Connectivity patterns in loop percolation I: the rationality phenomenon and constant term identities