Spatial Coalescent Models

The proposal is an investigation into models involving spatial coalescence, with the broad aim of gaining an understanding of the evolutions of models involving these mechanisms, for example in their dynamics or stationary distributions, but also, especially in applied settings, for the universal features of their spatial and temporal statistics, for example via the derivation of concrete scaling laws for associated probabilities and expectations in terms of the model parameters.

In particular to gain an understanding of:

I. Large time density estimates for multi-species coalescence models;

II. Negative correlations properties for models with coalescence;

III. Pfaffian, and extended Pfaffian, properties for the particle densities, and joint particle-mass densities, for spatial models with coalescence;

IV. Steady states for coalescent models with immigration;

V. The evolution for the eigenvalues for the real Ginibre matrices and its connection with annihilating systems.

The national impact of the work in this proposal can be summarized as:

1. Strengthening of the knowledge base within the field of particle systems for problems involving spatial coalescence, putting the UK at the cutting edge in this active area.

2. Creation of a UK group investigating an interlocking set of tools and applications in this area.

3. Training of young mathematicians for their future careers in science.

4. A contribution to the UK's provision for applied probability by targeting some of the group towards applications ranging from aggregation models in astronomy to interacting financial agents.