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

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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.