论文标题
短程和远程顺序：HOM移位中的块状行为过渡
Short-range and long-range order: a transition in block-gluing behavior in Hom shifts
论文作者
论文摘要
储层计算是预测湍流的有力工具，其简单的架构具有处理大型系统的计算效率。然而，其实现通常需要完整的状态向量测量和系统非线性知识。我们使用非线性投影函数将系统测量扩展到高维空间，然后将其输入到储层中以获得预测。我们展示了这种储层计算网络在时空混沌系统上的应用，该系统模拟了湍流的若干特征。我们表明，使用径向基函数作为非线性投影器，即使只有部分观测并且不知道控制方程，也能稳健地捕捉复杂的系统非线性。最后，我们表明，当测量稀疏、不完整且带有噪声，甚至控制方程变得不准确时，我们的网络仍然可以产生相当准确的预测，从而为实际湍流系统的无模型预测铺平了道路。
Hom shifts form a class of multidimensional shifts of finite type (SFT) and consist of colorings of the grid Z2 where adjacent colours must be neighbors in a fixed finite undirected simple graph G. This class includes several important statistical physics models such as the hard square model. The gluing gap measures how far any two square patterns of size n can be glued, which can be seen as a measure of the range of order, and affects the possibility to compute the entropy (or free energy per site) of a shift. This motivates a study of the possible behaviors of the gluing gap. The class of Hom shifts has the interest that mixing type properties can be formulated in terms of algebraic graph theory, which has received a lot of attention recently. Improving some former work of N. Chandgotia and B. Marcus, we prove that the gluing gap either depends linearly on n or is dominated by log(n). We also find a Hom shift with gap Θ(log(n)), infirming a conjecture formulated by R. Pavlov and M. Schraudner. The physical interest of these results is to better understand the transition from short-range to long-range order (respectively sublogarithmic and linear gluing gap), which is reflected in whether some parameter, the square cover, is finite or infinite.
