We present here a brief summary of results on so-called ‘‘exotic’’ directed lattices having nonregular periodicity. Such lattices, e.g., the Archimedean nets, are characterized by different site types and a spread of coordination numbers. The evidence adduced here shows that the exponent structure for the growth of animals and trees on such lattices is of the form predicted, with ${\mathrm{\theta }}_0$ for trees equivalent to θ for unrestricted animals, independent of the periodicity property. This supports the general form ${\mathrm{\theta }}_{\mathit{c}}$=${\mathrm{\theta }}_0$-c, for cycles c=0. We quote values for the growth parameter (or inverse critical fugacity) λ for bond trees and animal growth on selected directed lattices. The convergence of such series is well known to be subject to the influence of subdominant singularities in addition to θ, and we report on results obtained using the second-log-derivative scheme for the lattices of interest. Recent results from percolation studies for the alternating nets [H. J. Ruskin, Phys. Lett. A 162, 215 (1992)] have proved particularly encouraging for the Archimedean lattices, with z̃=2.498+-0.010, but uncertainties for the honeycomb were found to be large. Investigation of the site-to-bond ratios for animals and trees on the honeycomb gives somewhat smoother series behavior, which, though subject to confluence effects, supports a much lower value of the effective coordination number. We quote z̃=1.450+-0.050.

• Received 16 December 1991

DOI:https://doi.org/10.1103/PhysRevA.46.1797