Patterns generation and transition matrices in multi-dimensional lattice models

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