高維度網格型模型之花樣形成與置換矩陣

Abstract

此篇論文主要的目的是在高維度網格型模型中針對較多的符號較大的網格探討其花樣形成及置換矩陣的問題。利用在花樣上定義次序來得到次序矩陣和相對應的置換矩陣的遞迴公式，並且可藉由置換矩陣最大的特徵值來計算熵。

The aim of this paper is to study the pattern generation problems for more symbols on larger lattice with edge $2\ell$ in $d$-dimensional models, $d\geq 3$. Defining orderings for pattern $U$ on $\Sigma_{2\ell \times 2\ell \times \ldots \times 2\ell}$ on $\mathbf{Z}_{2\ell \times 2\ell \times \ldots \times 2\ell} \subset \mathbf{Z}^{d+1}$ enable us to derive simple recursion formulas for generating ordering matrices and the corresponding transition matrices. Furthermore, the spatial entropy can be computed through the maximum eigenvalue of transition matrices.