Decreasing sandpiles model the dynamics of configurations where each position $i\in \mathbb{N}$ contains a finite number of stacked grains $h_i$, such that $h_i\ge h_{i+1}$ (decrease property). Grains move according to a decreasing local rule $\mathcal{R}=(r_1,r_2,\dots ,r_p)$ such that $r_j\ge r_{j+1}$, meaning that $r_j$ grains move from columns i to $i+j$ for all $1\le j\le p$, if it does not contradict the decrease property. We are interested in the fixed point reached starting from a finite number of grains on a unique column.

In [21], we proved the emergence of wave patterns periodically covering fixed points, for rules of the form $(1,\dots ,1)$ (Kadanoff sandpile models). The present work is a significative extension: for large classes of decreasing sandpile model instances, we prove the emergence of patterns of various shapes periodically covering fixed points. We introduce new automata to analyze their asymptotic structure, and use the least action principle. The difficulty of understanding the behavior of sandpile models, despite the simplicity of the rules, is what makes the problem challenging.

减少的沙堆可以模拟每个位置的配置动态 $\mathrm{一世}\in \mathbb{ñ}$ 包含有限数量的堆积颗粒 $H_{\mathrm{一世}}$，这样 $H_{\mathrm{一世}}\ge H_{\mathrm{一世}+\mathrm{1个}}$（减少财产）。谷物按照递减的当地规则移动$\mathcal{[R}=（{\mathrm{[R}}_{\mathrm{1个}}，{\mathrm{[R}}_2，\dots ，{\mathrm{[R}}_p）$ 这样 ${\mathrm{[R}}_{Ĵ}\ge {\mathrm{[R}}_{Ĵ+\mathrm{1个}}$， 意思是 ${\mathrm{[R}}_{Ĵ}$谷物从第i列移至$\mathrm{一世}+Ĵ$ 对所有人 $\mathrm{1个}\le Ĵ\le p$，如果不与减少属性矛盾。我们感兴趣的是从唯一列上有限数量的晶粒开始达到的固定点。

在[21]中，对于形式规则，我们证明了周期性覆盖固定点的波动模式的出现 $（\mathrm{1个}，\dots ，\mathrm{1个}）$（Kadanoff沙堆模型）。当前的工作是一个有意义的扩展：对于大量递减的沙堆模型实例，我们证明了周期性覆盖固定点的各种形状的图案的出现。我们引入新的自动机来分析其渐近结构，并使用最小作用原理。尽管规则简单，但难以理解沙堆模型的行为，这使该问题具有挑战性。