Crossover from Isotropic to Directed Percolation

TL;DR

This paper introduces a generalized percolation model that interpolates between isotropic and directed percolation, analyzing its critical behavior and thresholds through extensive simulations.

Contribution

It proposes the biased directed percolation (BDP) model with anisotropic probabilities, extending the understanding of percolation transitions between isotropic and directed cases.

Findings

Percolation thresholds for p_d=0.6 and 0.8 were located.

Critical exponents match those of standard directed percolation.

The asymmetric perturbation near isotropic percolation is relevant and characterized.

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 $p_{\downarrow}= p \cdot p_d$ and $p_{\uparrow}=p \cdot (1-p_d)$, with $p $ representing the average occupation probability and $p_d$ 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 {\em biased directed percolation} (BDP). Standard isotropic percolation (IP) and DP are the two limiting cases of the BDP model, corresponding to $p_d=1/2$ and $p_d=0,1$ respectively. In this work, besides IP and DP, we also consider the $1/2<p_d<1$ 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 $p_d=0.6$ 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 $p_d -1/2 \neq 0$ near IP, and confirm that such an asymmetric scaling field is relevant at IP.