Abelian沙堆模型的随机变体

在Abelian沙堆模型中，沙粒随机添加到图的顶点上，当顶点处的沙粒数量达到某个阈值（通常是该顶点的度数）时，该顶点将变得不稳定。 不稳定的顶点坍塌，使得沙粒流向图上邻近顶点。 一个特殊的顶点（称为汇sink）吸收了系统中多余的沙粒，因此该过程最终趋于稳定。.这个模型最初由Bak等人于1987年提出，并由Dhar于1990年加以概括和正式化。从那以后，它在不同的领域引起了极大的兴趣——数学、计算机科学、统计物理学——应用领域包括地震、脑细胞放电和森林火灾等。.该项目旨在研究Abelian沙堆模型的随机变体，在其不稳定顶点的坍塌中引入了额外一个随机性。不稳定顶点选择邻近顶点的一个随机子集发送一个或多个沙粒，忽略其他邻近顶点，而不是向所有邻近顶点发送沙粒。相关文献中有若干关于这样的随机沙堆模型的论述，但是关于它们的理论结论并不多。 该项目旨在填补这一知识缺口。

In the Abelian sandpile model, grains of sand are added at random to vertices on a graph. When the number of grains at a vertex reaches a certain threshold (typically the degree of that vertex), that site becomes unstable. Unstable sites topple, sending one grain of sand to each of their neighbours on the graph. A special vertex, called the sink, absorbs excess grains from the system, and so the process eventually stabilises..This model was originally introduced in 1987 by Bak et al., and generalised and formalised by Dhar in 1990. Since then, it has attracted considerable interest in different fields - mathematics, computer science, statistical physics - with applications in areas such as earthquakes, the firings of brain cells, and forest fires..This project aims to study stochastic variants of the Abelian sandpile model, where an extra layer of randomness is introduced into the topplings of unstable vertices. Instead of sending grains to each of its neighbours, an unstable vertex chooses a random subset of these neighbours to send one or more grains to, ignoring the others. A number of such stochastic sandpile models exist in the literature, but not many theoretical results are known about these. This project seeks to bridge this knowledge gap.