Patterns generation and spatial entropy in two-dimensional lattice models

Abstract

Patterns generation problems in two-dimensional lattice models are studied. Let S be the set of p symbols and ℤ2ℓ×2ℓ, ℓ ≥ 1, be a fixed finite square sublattice of ℤ2. Function U: ℤ 2ℓ×2ℓ → S is called local pattern. Given a basic set B of local patterns, a unique transition matrix A2 which is a q2 × q2 matrix, q = pℓ2, can be defined. The recursive formulae of higher transition matrix An on ℤ 2ℓ×nℓ have already been derived. Now Anm, m ≥ 1, contains all admissible patterns on ℤ(m+1)ℓ×nℓ which can be generated by B. In this paper, the connecting operator ℂ m, which comprises all admissible patterns on ℤ (m+1)ℓ×2ℓ, is care fully arranged. ℂm can be used to extend Anm to Anm+1 recursively for n ≥ 2. Furthermore, the lower bound of spatial entropy h(A2) can be derived through the diagonal part of ℂ m. This yields a powerful method for verifying the positivity of spatial entropy which is important in examining the complexity of the set of admissible global patterns. The trace operator Tm of ℂm can also be introduced. In the case of symmetric A2, T2m gives a good estimate of the upper bound on spatial entropy. Combining ℂm with Tm helps to understand the patterns generation problems more systematically