λ > 4

Abstract

The universal constant λ , the growth constant of polyominoes (think Tetris pieces), is rigorously proved to be greater than 4.

Ferromagnetic Potts models with multisite interaction

“…On the other hand, Fig. 5 remarkably confirms (18), suggesting a divergence of the corre-lation length ξ(4) ∝ |t| −ν as t → 0. The indefiniteness of the four-state model manifested both analytically and numerically, is in agreement with renormaliztion group (RG) predictions.…”

“…As clearly shown in Fig. 5, the specific heat, normalized by L α/ν , collapses on a single curve as follows from (18). Thus, it is reasonable to assume the hyperscaling relation indeed holds, in consistency with the scaling relations we use.…”

“…where G(k, m k ) is the number of configurations with k faces and m k sites, associated with a k cluster. It is known [18][19][20] that the combinatorial term g k = m k G(k, m k ) for large k is the asymptotic number of lattice animals [21] g k ≈ cλ k /k, where λ ≈ 4.0626 and c ≈ 0.3169. This observation distinguishes between q > 4 and q ≤ 4.…”