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: dcI=6 for isotropic, and
dcD=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.Comment: 19 pages, written in RevTeX, 12 figures available upon request (from
[email protected] or [email protected]), EF/UCT--94/2, to be
published in Phys. Rev. E (May 1994