Processes of coalescence and fragmentation: phase transitions, scaling limits and self-organised criticality

这个项目集中于涉及合并和分裂现象的丰富的概率模型集合。我们提出了一种统一的方法来研究在不同环境中分别引起显著关注的一大类模型。一个特别的目标是给出新的和强大的标度极限的描述，并理解相变和自组织临界现象。这些主题在物理学、概率论、离散数学和复杂系统的广泛领域都具有相当重要的意义。一些被研究的系统还将作为分析某些空间模型的工具，由于它们与量子自旋系统和超流等物理主题的关系而具有活跃的当前兴趣。该项目的最后一部分集中在随机交错，这是一种新兴的渗流和侵蚀模型，长程相互作用在其中扮演着重要的角色。我们将结合一系列概率、几何和分析工具来描述出现的离散随机结构，并了解它们的性质与传统渗流模型的性质有何不同。

This project is focused on a rich collection of probabilistic modelswhich involve phenomena of coalescence and fragmentation.We propose a unified approach to the study of a large class of modelswhich have separately attracted significant attention in a variety ofcontexts.A particular goal is to give new and powerful descriptions of scalinglimits, and to understand phenomena of phase transition andself-organised criticality. Such themes are of considerablesignificance across a wide range of fields in physics, probabilitytheory, discrete mathematics and complex systems.Some of the systems studied will also serve as tools in the analysisof certain spatial models, which are of lively current interest due totheir relationship to quantum spin systems and physical topics such assuperfluidity.The last part of the project centres on random interlacements, anemerging model of percolation and erosion in which long-rangeinteractions play a significant role. We will combine a range ofprobabilistic, geometric and analytic tools to describe the discreterandom structures that emerge, and to understand how their propertiesdiffer from those of traditional percolation models.