Percolation with restricted valence

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Abstract
The authors have enumerated exactly the square lattice site animals with vertex degree less than or equal to three and with up to fourteen sites in the cluster. Using series analysis techniques estimates have been made of the location, exponent and amplitude of the principal singularity in the animal generating function for maximum vertex degree v=2, 3 and 4. It is shown rigorously that the dominant singularity in the generating function must have a different exponent for v=2 than for v=3 and the numerical results suggest that the generating function diverges with an exponent of 4/3 for v=2 but logarithmically for both v=3 and v=4.

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