The histogram characteristics of perimeter polynomials for directed percolation

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Abstract
New perimeter polynomials (in dimensions d = 2 to 4) are analysed for directed site percolation. A study of these data shows that i) above pc the average perimeter-to-size- ratio varies as α = (1 — p) /p + Bs-1/d; ii) At pc its leading correction term estimates supports the prediction (from scaling) of an exponent equal to 1/Δ — 1 (with Δ the gap exponent for directed percolation); iii) At p = 0 the limiting ratio is estimated on various lattices. Fairly definitive evidence is obtained in favour of α(p = 0) = 3/4 for the square site animals and this result is used to study the second correction term which is estimated to be analytic (∼ s-2) as the first correction term (Bethe — like and∼ s-1, without any obvious dimensional dependence)