Percolation clusters are probably the simplest example for scale-invariant structures which are either 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 unresolved problem because it is accompanied by different upper critical dimensions: ${\mathit{d}}_{\mathit{c}}^{\mathrm{I}}$=6 for isotropic and ${\mathit{d}}_{\mathit{c}}^{\mathrm{D}}$=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 effecive 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.

• Received 3 December 1993

DOI:https://doi.org/10.1103/PhysRevE.49.5058