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 Z 22 , 1, be a fixed finite square sublattice of Z 2 . Function U : Z 22 S is called local pattern. Given a basic set B of local patterns, a unique transition matrix A 2 which is a q 2 q 2 matrix, q = p 2 , can be defined. The recursive formulae of higher transition matrix An on Z 2n have already been derived Now A m n , m 1, contains all admissible patterns on Z (m+1)n which can be generated by B. In this paper, the connecting operator Cm, which comprises all admissible patterns on Z (m+1)2 , is carefully arranged. Cm can be used to extend A m n to A m n+1 recursively for n 2. Furthermore, the lower bound of spatial entropy h(A 2 ) can be derived through the diagonal part of Cm. 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 A 2 , T 2m gives a good estimate of the upper bound on spatial entropy. Combining Cm with Tm helps to understand the patterns generation problems more systematically.