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Updated: Aug 16, 2026

Evolution of Staircase Structures in Diffusive Convection
Published on: September 5, 2018
Inverse cascade in a percolation model: hierarchical description of time-dependent scaling
Ilya Zaliapin1, Henry Hang Lam Wong, Andrei Gabrielov
1Institute of Geophysics and Planetary Physics, University of California, Los Angeles, California 90095, USA. zal@ess.ucla.edu
Abstract:
The dynamics of a two-dimensional site percolation model on a square lattice is studied using the hierarchical approach introduced by Gabrielov [Phys. Rev. E 60, 5293 (1999)]. The key elements of the approach are the tree representation of clusters and their coalescence, and the Horton-Strahler scheme for cluster ranking. Accordingly, the evolution of the percolation model is considered as a hierarchical inverse cascade of cluster aggregation. A three-exponent time-dependent scaling for the cluster rank distribution is derived using the Tokunaga branching constraint and classical results on percolation in terms of cluster masses. Deviations from the pure scaling are described. An empirical constraint on the dynamics of a rank population is reported based on numerical simulations.
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