Crossover from Isotropic to Directed Percolation

TL;DR

This paper investigates the transition from isotropic to directed percolation, using renormalization group theory to analyze crossover behavior and predict the scale at which anisotropic effects dominate.

Contribution

It introduces a consistent renormalization group approach to study the crossover between isotropic and directed percolation, including the calculation of crossover and effective exponents.

Findings

Identified the crossover exponent between isotropic and directed percolation.

Calculated effective exponents for different length scales and pair correlation functions.

Predicted the anisotropy scale where the crossover occurs.

Abstract

Percolation clusters are probably the simplest example for scale--invariant structures which either are governed by isotropic scaling--laws (``self--similarity'') or --- as in the case of directed percolation --- may display anisotropic scaling behavior (``self--affinity''). Taking advantage of the fact that both isotropic and directed bond percolation (with one preferred direction) may be mapped onto corresponding variants of (Reggeon) field theory, we discuss the crossover between self--similar and self--affine scaling. This has been a long--standing and yet unsolved problem because it is accompanied by different upper critical dimensions: $d_c^{\rm I} = 6$ for isotropic, and $d_c^{\rm D} = 5$ for directed percolation, respectively. Using a generalized subtraction scheme we show that this crossover may nevertheless be treated consistently within the framework of renormalization group theory. We identify the corresponding crossover exponent, and calculate effective exponents for different length scales and the pair correlation function to one--loop order. Thus we are able to predict at which characteristic anisotropy scale the crossover should occur. The results are subject to direct tests by both computer simulations and experiment. We emphasize the broad range of applicability of the proposed method.