2009/07/05 by A. Nihat Berker, Michael Hinczewski, Roland R. Netz · 1 citation
Mathematics · Physics and Astronomy · #Combinatorics #Complex Network Analysis Techniques #Composite material #Condensed matter physics #Critical exponent #Critical phenomena #Directed percolation #Electrical resistivity and conductivity #Ising model #Kosterlitz–Thouless transition #Materials science #Mathematical physics #Mathematics #Opinion Dynamics and Social Influence #Percolation (cognitive psychology) #Percolation critical exponents #Percolation threshold #Phase (matter) #Phase diagram #Phase transition #Physics #Quantum mechanics #Range (aeronautics) #Renormalization group #Scale (ratio) #Statistical physics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Topology (electrical circuits) #cond-mat.dis-nn #cond-mat.stat-mech
paper · pdf · doi:10.1103/physreve.80.041118
published as Phys. Rev. E 80, 041118 (2009) · Added explanations and data. Published version. 4pages, 4 figures
openalex publication_date 2009/10/15 · arxiv created 2009/12/14 · arxiv updated 2010/01/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
Percolation in a scale-free hierarchical network is solved exactly by renormalization-group theory in terms of the different probabilities of short-range and long-range bonds. A phase of critical percolation, with algebraic [Berezinskii-Kosterlitz-Thouless (BKT)] geometric order, occurs in the phase diagram in addition to the ordinary (compact) percolating phase and the nonpercolating phase. It is found that no connection exists between, on the one hand, the onset of this geometric BKT behavior and, on the other hand, the onsets of the highly clustered small-world character of the network and of the thermal BKT transition of the Ising model on this network. Nevertheless, both geometric and thermal BKT behaviors have inverted characters, occurring where disorder is expected, namely, at low bond probability and high temperature, respectively. This may be a general property of long-range networks.