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Essential Singularity in a Model of Branched Polymers

We study the asymptotic properties of large branched polymers (lattice animals) on a fractal lattice, the two-dimensional modified Sierpinski gasket of base b = 3. The generating function displays an essential singularity of the form G(x) ∼ exp [c(xc - x)-ψ], with ψ = ln(3 - √3)/ln(3 + √3) ≃ 0.1527. This is in contrast to the power law singularity found for periodic lattices and some other fractal lattices such as the standard Sierpinski gasket. This result suggests that similar essential singularities may also exist for polymers in nonhomogeneous media.