Directed Forest Fire Models

平均场森林火灾模型是由拉特和托特（EJP，2009）引入的一个抽象的数学模型，通过严格的数学方法来研究自组织临界（SOC）现象。这是一个不平衡的随机分裂-合并过程的例子。它是完全图Kn的一个演化的随机子图，其中每个可能的无向边以n-1的速率独立地添加到图中，就像在动态Erdos-Renyi图中一样，但是连通分量以与它们的顶点数成比例的速率随机删除。这些被称为火灾的删除是由闪电触发的，闪电以n-1/2的速率独立地击中每个顶点。已知大n的极限行为表现出SOC，在这个意义上，连接组件大小的极限分布在有限时间内变得至关重要，并且此后保持关键：它有一个多项式衰减的尾部，与我们在关键的鄂尔多斯看到的指数相同这个博士研究项目旨在将上述模型的已知定理扩展到一个变体，其中边缘是定向的。这意味着火只能沿沿着每个边在指定方向上传播。当闪电击中一个顶点v时，v会燃烧，而从v开始的有向路径在那一刻可能到达的每个顶点w也会燃烧。为了研究有向林火动力学，必须研究n个顶点的入图和出图的重叠结构。科学的意图是将该模型的定性适用性扩展到现实生活中的SOC系统，这些系统的各个元素之间具有方向或因果关系，例如大脑和金融网络中的神经元级联。 该项目的第一个目标是证明这个有向模型在上述意义上显示SOC，当查看入图和出图大小而不是连接组件大小时。这是可能的，以证明这一限制类的初始条件，使用耦合参数，涉及顶点生存时间的有向和无向的过程。第一个技术任务是将其扩展到更一般的初始条件，使用拉特和托特的鞅近似参数来证明有向平均场森林火灾模型中的图内大小的经验分布具有解决临界森林火灾方程的极限。 我们期望图外大小的经验分布的演化比图内的演化复杂得多。我们期望它的极限动力学不能自主描述，而是必须在一些更大的标记有根树空间上使用分布。该项目的第二个目标是证明一个极限定理，描述当n趋于无穷大时外图的动态。 第三个目标将是研究标记顶点的出图的动力学。在初始条件是年龄驱动的非齐次随机图的特殊情况下，出图将被近似为一个多类型的Galton-Watson树，它经历了生长和修剪事件。这一过程将在时间平稳的设置类型的分布是唯一的不动点的年龄分布PDE的研究。一个具有挑战性的问题将是决定一个标记的顶点的出图的局部极限过程是爆炸性的或几乎肯定保持有限的所有时间。计划开始研究一些简单的动态随机分枝树，这些树同时具有生长和修剪事件。本项目福尔斯属于EPSRC数学科学研究领域，特别是以下主题：应用概率和统计

The mean field forest fire model is an abstract mathematical model introduced by Rath and Toth (EJP, 2009) to study the phenomenon of self-organized criticality (SOC) by a rigorous mathematical approach. It is an example of a random fragmentation-coalescence process out of equilibrium. It is an evolving random subgraph of the complete graph Kn, in which each possible undirected edge is added to the graph rate n-1, independently, as in the dynamical Erdos-Renyi graph, but connected components are randomly deleted at a rate proportional to their number of vertices. The deletions, called fires, are triggered by a lightning strikes, hitting each vertex independently at rate n-1/2. The limiting behaviour for large n is known to exhibit SOC, in the sense that the limiting distribution of connected component sizes becomes critical in a finite time and remains critical thereafter: it has a polynomially decaying tail, with the same exponent as one sees in the critical Erdos-Renyi graph and in critical branching processes.This PhD research project aims to extend known theorems about the above model to a variant in which the edges are directed. This means that fire can only propagate in a specified direction along each edge. When lightning strikes a vertex v, then v burns and so does every vertex w that may be reached at that moment by a directed path from v. Each fire causes all edges incident on any burned vertex to be deleted instantaneously. To study the directed forest fire dynamics, one must study the overlapping structures of in-graphs and out-graphs of the n vertices. The scientific intention is to extend the qualitative applicability of the model to real-life SOC systems which have directional or causal relationships between their individual elements, such as the neuronal cascades in the brain and financial networks. The first goal of the project is to prove that this directed model displays SOC in the above sense, when looking at in-graph and out-graph sizes rather than connected component sizes. It is possible to prove this for a restricted class of initial conditions using a coupling argument that relates vertex survival times in the directed and undirected processes. The first technical task is to extend this to more general initial conditions, using the martingale approximation arguments of Rath and Toth to prove that the empirical distribution of in-graph sizes in the directed mean field forest fire model has a limit that solves the critical forest fire equations. We expect the evolution of the empirical distribution of out-graph sizes to be much more complicated than the in-graph evolution. We expect that its limiting dynamics cannot be described autonomously, but instead one must work with a distribution on some larger space of labelled rooted trees. The second goal of the project will be to prove a limit theorem describing the dynamics of the out-graphs as n tends to infinity. The third goal will be to study the dynamics of the out-graph of a tagged vertex. In the special case where the initial condition is an age-driven inhomogeneous random graph, the out-graph will be approximated by a multitype Galton-Watson tree, which experiences both growth and pruning events. This process will be studied in the time-stationary setting where the distribution of types is the unique fixed point of the age distribution PDE. A challenging question will be to decide whether the local limit process of the out-graph of a tagged vertex is explosive or almost surely remains finite for all time. It is planned to begin by studying some simpler dynamic random branching trees that have both growth and pruning events.This project falls within the EPSRC Mathematical Sciences research area, specifically the following themes: applied probability and statis