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Critical condition of the water-retention model

Seung Ki Baek1 and Beom Jun Kim2,*

  • 1Integrated Science Laboratory, Umeå University, 901 87 Umeå, Sweden†
  • 2BK21 Physics Research Division and Department of Physics, Sungkyunkwan University, Suwon 440-746, Korea

  • *beomjun@skku.edu
  • †Present address: School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea.
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Phys. Rev. E 85, 032103 – Published 29 March, 2012

DOI: https://doi.org/10.1103/PhysRevE.85.032103

Abstract

We study how much water can be retained without leaking through boundaries when each unit square of a two-dimensional lattice is randomly assigned a block of unit bottom area but with different heights from zero to n−1. As more blocks are put into the system, there exists a phase transition beyond which the system retains a macroscopic volume of water. We locate the critical points and verify that the criticality belongs to the two-dimensional percolation universality class. If the height distribution can be approximated as continuous for large n, the system is always close to a critical point and the fraction of the area below the resulting water level is given by the percolation threshold. This provides a universal upper bound of areas that can be covered by water in a random landscape.

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Phys. Rev. E 85, 032103– Published 29 March, 2012
  • Received 1 November 2011

©2012 American Physical Society

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