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

Percolation in a scale-free hierarchical network is solved exactly byrenormalization-group theory, in terms of the different probabilities ofshort-range and long-range bonds. A phase of critical percolation, withalgebraic (Berezinskii-Kosterlitz-Thouless) geometric order, occurs in thephase diagram, in addition to the ordinary (compact) percolating phase and thenon-percolating phase. It is found that no connection exists between, on theone hand, the onset of this geometric BKT behavior and, on the other hand, theonsets of the highly clustered small-world character of the network and of thethermal BKT transition of the Ising model on this network. Nevertheless, bothgeometric and thermal BKT behaviors have inverted characters, occurring wheredisorder is expected, namely at low bond probability and high temperature,respectively. This may be a general property of long-range networks.

Critical Percolation Phase and Thermal BKT Transition in a Scale-Free Network with Short-Range and Long-Range Random Bonds