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 propose a family of bipartite hierarchical lattice of order  N  governed by a pair of parameters  ℓ  and  γ . We study long-range percolation on the bipartite hierarchical lattice where any edge (running between vertices of unlike bipartition sets) of length  k  is present with probability  p    k   = 1 − exp(− αβ   − k  ), independently of all other edges. The parameter  α  is the percolation parameter, while  β  describes the long-range nature of the model. The model exhibits a nontrivial phase transition in the sense that a critical value  α    c   ∈ (0,  ∞ ) if and only if  ℓ  ≥ 1, 1 ≤  γ  ≤  N  − 1, and  β  ∈ ( N ,  N   2 ). Moreover, the infinite component is unique when  α  >  α    c  .

Phase Transition in Long-Range Percolation on Bipartite Hierarchical Lattices