Directed Forest Fire Models

平均场森林火灾模型是Rath和Toth(EJP，2009)提出的用严格的数学方法研究自组织临界性(SOC)现象的抽象数学模型。这是一个不平衡的随机破碎-合并过程的例子。它是完全图Kn的一个演化随机子图，其中每个可能的无向边被独立地添加到图速率n-1，就像在动态Erdos-Renyi图中一样，但连通分量以与其顶点数成比例的速率被随机删除。这种删除被称为火灾，是由闪电以n-1/2的速度独立击中每个顶点而触发的。大n的极限行为是已知的SOC，在这个意义上，连通分量大小的极限分布在有限时间内变得临界，此后仍然是临界的：它有一个多项式衰减的尾部，其指数与临界Erdos-Renyi图和临界分支过程中的指数相同。本博士研究项目的目的是将关于上述模型的已知定理推广到边有向的变体。这意味着火只能沿每条边沿指定方向传播。当闪电击中顶点v时，v被烧毁，此时从v开始的有向路径可能到达的每个顶点w也被烧毁。每一次火灾都会导致入射到任何被烧毁的顶点上的所有边瞬间被删除。要研究有向森林火灾动力学，必须研究n个顶点的入图和出图的重叠结构。其科学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性或因果关系，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，当查看图内和图外的大小而不是连接的组件大小时，该有向模型在上面的意义上显示SOC。对于一类受限的初始条件，可以用一个耦合参数来证明这一点，该参数与有向和无向过程中的顶点生存时间有关。第一个技术任务是将其推广到更一般的初始条件，使用Rath和Toth的鞅近似参数来证明有向平均场森林火灾模型中的图大小的经验分布存在解临界森林火灾方程的极限。我们预计图外尺寸的经验分布的演化将比图内的演化复杂得多。我们期望它的极限动力学不能被自主地描述，相反，人们必须在一些更大的标记有根的树的空间上进行分布。该项目的第二个目标将是证明一个极限定理，该极限定理描述了当n趋于无穷大时输出图的动力学。第三个目标是研究带标记的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，输出图将由经历增长和剪枝事件的多类型Galton-Watson树来逼近。这一过程将在时间固定的环境中研究，其中类型的分布是年龄分布PDE的唯一固定点。一个具有挑战性的问题将是确定标记顶点的外图的局部极限过程是爆炸性的还是几乎肯定总是有限的。这个项目属于EPSRC数学科学研究领域，特别是以下主题：应用概率和统计量

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