Patterns generation and transition matrices in multi-dimensional lattice models

Abstract

In this paper we develop a general approach for investigating pattern generation problems in multi-dimensional lattice models. Let S be a set of p symbols or colors, Z(N) a fixed finite rectangular sublattice of Z(d), d >= 1 and N a d-tuple of positive integers. Functions U : Z(d) --> S and U-N : Z(N) --> S are called a global pattern and a local pattern on ZN, respectively. We introduce an ordering matrix X-N for Sigma(N), the set of all local patterns on Z(N). For a larger finite lattice Z((N) over tilde) (N) over tilde >= N, we derive a recursion formula to obtain the ordering matrix X-(N) over tilde of Sigma((N) over tilde) from XN. For a given basic admissible local patterns set 13 C Ely, the transition matrix T-N(B) is defined. For each (N) over tilde >= N denoted by Sigma((N) over tilde)(B) the set of all local patterns which can be generated from B, the cardinal number of Sigma((N) over tilde)(B) is the sum of entries of the transition matrix T-(N) over tilde(B) which can be obtained from T-N(B) recursively. The spatial entropy h(B) can be obtained by computing the maximum eigenvalues of a sequence of transition matrices T-n(B). The results can be applied to study the set of global stationary solutions in various Lattice Dynamical Systems and Cellular Neural Networks.