New combinatorial perspectives on the abelian sandpile model

阿贝尔沙堆模型是一个动力学系统，出现在80年代后期，作为展示自组织临界概念的工具。粗略地说，这种自组织临界性的概念意味着一个系统朝着临界状态发展，当被轻推时，会发生所有距离和时间尺度的雪崩。原型沙堆模型是在平面网格上，但首选的数学设置是在图形上。这个模型的核心是它的倾倒动力学：如果沙堆长得太高，那么沙堆就会倾倒，并通过向相邻的沙堆捐赠沙粒来实现。这些相邻的沙堆可能会自己倒塌，这个过程会一直持续下去，直到系统达到某种稳定状态。虽然它已经被证明是一个糟糕的模型来模拟一般的沙堆，但它已经被证明是一个很好的模型来模拟许多其他和更重要的事情。例子很多，包括森林火灾，社交媒体，甚至毒理学中的剂量反应分析。该模型还解释了在这些系统中观察到的级联效应。随着数学家们对抽象图上沙堆模型的研究，出现了许多丰富的结果，这些研究也为数学的许多其他部分提供了联系。最近，作者对完全二部图上的沙堆模型进行了深入的研究，发现了一些新的令人惊讶的结果。一个这样的结果是，循环状态（类似于临界状态）可以唯一地表示为阶梯式多角形，几何对象就像多米诺骨牌，有许多单元格，但被封闭在两个阶梯形状之间。这一观察导致了一个新的链接多项式定义在这些polyominoes和对角调和多项式的代数组合，代数combinatorialists在过去的十年中更肥沃的狩猎场之一的主题之间的联系。我们的建议是遵循这一成功的应用分析，以更一般的类图是定期或递归的某种方式。目的是对这些图上的沙堆模型的循环状态进行分类，并确定它们与哪些其他组合对象相关联。此外，我们将把最初的工作放在它的头上，以建立一个新的双射组合学工具，将一般格的平铺与沙堆模型的经常性状态联系起来。这将提供新的见解，理论的格子镶嵌，也未解决的问题，在这一广泛的领域。

The abelian sandpile model is a dynamical system that appeared in the late eighties as the vehicle to showcase the concept of self-organised criticality. Roughly speaking, this concept of self-organised criticality means that a system evolves towards critical states that, when nudged, topple and cause avalanches of all distances and time scales to happen throughout.The prototypical sandpile model is on the planar grid but the preferred mathematical setting is on a graph. At the heart of this model are its toppling dynamics: if a sandpile grows too high then the pile topples and does so by donating grains of sand to its neighbouring piles. These neighbouring piles may themselves topple, and the process continues until the system reaches some stable state.Although it has been shown to be a poor model for modelling general sandpiles, it has been shown to be a good model for many other and more important things. Examples are plentiful and include forest fires, social media, and even dose response analysis in toxicology. The model also explains the cascading effects that have been observed in these systems. Many rich results emerged when mathematicians began to study the sandpile model on abstract graphs and these studies have also provided links to many other parts of mathematics.Very recently, the author conducted an in-depth study of the sandpile model on the complete bipartite graph, unearthing new and surprising results. One such result is that recurrent states (similar to critical states) can be uniquely represented as staircase polyominoes, geometric objects that are like dominoes with many cells but which are enclosed between two staircase shapes. This observation led to a new link between polynomials defined on these polyominoes and the subject of diagonal harmonic polynomials in algebraic combinatorics, one of the more fertile hunting grounds for algebraic combinatorialists in the last decade.Our proposal is to follow the success of this by applying the analysis to more general classes of graphs that are regular or recursive in some way. The purpose is to perform a classification of recurrent states of the sandpile model on these graphs and determine what other combinatorial objects they are linked to. Further to this we will turn the initial work on its head to build a new tool in bijective combinatorics that will relate tilings of general lattices to recurrent states of the sandpile model. This will provide new insights into the theory of lattice tilings, and also unsolved problems in this broad area.