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 Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We have found analytical expressions (polynomials) of the percolationprobability for site percolation on a square lattice of size $L \times L$ siteswhen considering a plane (the crossing probability in a given direction), acylinder (spanning probability), and a torus (wrapping probability along onedirection). Since some polynomials are extremely cumbersome, they are presentedas separate files in Supplemental material. The system sizes for which this wasfeasible varied up to $L=17$ for a plane, up to $L=16$ for a cylinder, and upto $L=12$ for a torus. To obtain a percolation probability polynomial, allpossible combinations of occupied and empty sites have to be taken intoaccount. However, using dynamic programming along with some ideas related tothe topology, we offer an algorithm which allows a significant reduction in thenumber of configurations requiring consideration. A rigorous formal descriptionof the algorithm is presented. Divisibility properties of the polynomials havebeen rigorously proved. Reliability of the polynomials obtained have beenconfirmed by the divisibility tests. The wrapping probability polynomials on atorus provide a better estimate of the percolation threshold than that from thespanning probability polynomials. Surprisingly, even a naive finite sizescaling analysis allows an estimate to be obtained of the percolation threshold$p_c = 0.59269$.

Exact percolation probabilities for a square lattice: Site percolation on a plane, cylinder, and torus