Self-organized criticality and asymptotic freedom in random forests

Polymers are assemblies of large numbers of molecules linked together by chemical bonds. From a theoretical perspective it is natural to ask if the properties of systems of polymers can be predicted from first principles. That is, if you are told the properties of the molecules, can you deduce the properties of the polymers the molecules can form? Solving this problem with full physical fidelity is extremely difficult. Fortunately, for many important questions, simplified mathematical models produce meaningful and practically useful answers. One type of question that can be asked about polymers concerns the existence of a gelation transition. This transition is familiar from everyday life: when preparing a gelatine dessert, hot water is combined with a concentrated form of gelatine. The result is a liquid that, once cooled, sets into a firm gel. Physically, at high temperatures there are few chemical bonds, and as a result, the molecules form only small polymers that disperse throughout the water. At low temperatures chemical bonds form more readily, and a dense network of them produce a global structure: the gel. This proposal is about a model for the gelation transition of branched polymers, which occur when the molecular structure prevents chemical bonds from forming cycles. The constraint defining branched polymers leads to many fascinating connections. Namely, questions about branched polymers can be reformulated into questions about a mathematically well-formulated quantum field theory. Quantum field theory is a notoriously challenging subject from the mathematical point of view. Recently, however, it has been realised that it is possible to utilise the field theory perspective to show that branched polymers cannot form a very thin layer of gel: there is no gelation transition in two dimensions. There is, however, a gelation transition in three dimensions! Many questions about branched polymers remain unanswered, closely mirroring our mathematically incomplete understanding of quantum field theory. The first goal of this proposal is to develop our understanding of the absence of a gelation transition in two dimensions. This is known as asymptotic freedom, an important feature of our most fundamental physical theories. A thorough mathematical understanding is lacking, and this proposal will make progress by developing our understanding in the branched polymer context. The second goal of this proposal is to understand the fact that branched polymers exhibit self-organised criticality when a gelation transition does occur. While this is understood in specific contexts from a probabilistic point of view, it is not understood from the field theory perspective. The study of branched polymers is rich as it lies at the crossroads of many mathematical subjects, and hence branched polymers provide a concrete setting in which to develop tools of much wider applicability. In particular, this proposal will develop methods for studying coagulation-fragmentation dynamics as well as rigorous renormalisation group methods. These are topics of broad theoretical and applied interest, and it is expected that their development will lead to future applications in probability theory and theoretical computer science.