The critical group of a topological graph: an approach through delta-matroid theory

沙堆模型是一个在物理学中被广泛研究的模型，并在经济学中用于模拟当其中一家银行违约时银行网络的后果。它有一个简单直观的描述如下。假设我们把沙粒堆在一些地方。在某个时候，一堆沙子变得太大，无法保持直立，它会倒塌，雪崩发生，沙子会转移到相邻的地方。接下来会发生什么？转移到邻近地点的沙子数量是否足以使它们倾倒？这个过程要持续多久，直到所有的地点再次变得稳定，我们最终得到的沙粒是什么形状？这种情况背后的制度和模式是什么？现在假设我们有一个网络（例如，由计算机服务器或通过电缆连接的风力发电场组成）。在网络断开之前，必须有多少根电缆出现故障？如果每条电缆都有一定的故障概率，那么网络断开的概率是多少？这种情况背后的制度和模式是什么？事实证明，这两种情况和许多其他情况下的模式与称为“临界群”的代数结构密切相关。网络在数学中被称为“图”的对象建模。每一个图都有一个与之相关的临界群，临界群的产生有许多不同的方式，也有许多不同的数学和物理图的应用;例如，通过切屑燃烧或沙堆模型，流和割空间，计算生成树，甚至停车场的数学模型。到目前为止，在我们描述的情况下，没有几何学：重要的是相邻性，当另一个站点倒塌时，哪些站点会得到额外的沙子，以及哪些计算机对通过电缆连接。网络的外观对于我们描述的模型并不重要，但是，在许多情况下，图具有通过将其嵌入表面而提供的几何结构（例如，网络可能绘制在环面上），几何结构以及邻接关系决定了图的关键属性。例子包括模拟酶或某些形式的DNA链的作用的图表。最近的一些工作暗示了图的几何结构和临界群之间存在深层联系的可能性。当人们离开可以在平面上绘制的图时，与图相关的一些基本对象发生了深刻的变化：特别是研究生成准树而不是生成树。到目前为止，对嵌入曲面图的临界群的研究还没有考虑这些变化的基本结构。我们的目标是探索这个空间，通过发展一个考虑到嵌入的临界群理论来实现几何结构的全部潜力。

The Sandpile Model is a widely studied model in physics and used in economics to model the consequences for a network of banks when one of them defaults. It has a simple intuitive description as follows. Suppose we pile up grains of sand on a number of sites. At some point one pile becomes too large to stay upright, it topples, an avalanche occurs and sand gets transferred to adjacent sites. What happens next? Is the amount of sand transferred to the adjacent sites enough to make them topple? For how long does the process continue until all the sites become stable again and what configuration of grains of sand do we end up with? What are the systems and patterns underlying this situation? Now suppose we have a network (comprising, for example, computer servers or wind farms connected by cables). How many cables must fail before the network becomes disconnected? If each cable has a certain probability of failure, what is the probability that the network becomes disconnected?What are the systems and patterns underlying this situation?It turns out that the patterns underlying these two situations and many others are closely related to algebraic structures called 'Critical Groups'. Networks are modelled in mathematics by objects called 'graphs'. Each graph has a Critical Group associated with it. Critical Groups arise in many different ways and in many different applications of graphs in mathematics and physics; for example, through Chip-Firing or the Sandpile Model, Flow and Cut Spaces, counting spanning trees and even mathematical models of car parking. In the situations we have described so far there is no geometry: all that matters is adjacency and which sites receive extra sand when another topples and which pairs of computers are linked by a cable. The way that the networks look is not important for the models we have described.However, in many settings a graph comes equipped with geometric structure provided by an embedding of it in a surface (for example, a network may come drawn on a torus), and the geometric structure as well as the adjacencies determine the key properties of the graph. Examples include graphs that model the actions of enzymes or certain forms of DNA strands. Some recent work has hinted at the possibility that there are deep links between the geometric structure of graphs and Critical Groups. When one moves away from graphs that can be drawn on a plane, some of the fundamental objects associated with the graph change profoundly: specifically one studies spanning quasi-trees rather than spanning trees. So far, the study of Critical Groups for graphs embedded on surfaces has not taken into account these changed fundamental structures. We aim to explore this space, realising the full potential of the geometric structure by developing a theory of the Critical Group that takes into account the embedding.