Some contributions at the interface of combinatorial probability and continuum structures

The interplay between discrete and continuum structures in Probability has a long history dating back to early versions of the Central Limit Theorem in the 18th and 19th century as a way to describe large discrete structures by a continuum approximation. Over the course of the 20th century, there have been numerous extensions and abstractions of this notion including Brownian motion as limit of random walks, diffusion approximations to population models, continuum representations of large random graphs and continuum models in mean-field or lattice models of Statistical Physics. Such "scaling limits" are often universal in the sense that they capture the large-system behaviour of large classes of discrete models that differ in their local behaviour.
Of particular relevance for the present project are recent developments in the area of branching structures, specifically for underlying genealogical trees, as well as structures in which branching arises as a technique rather than an intrinsic feature of the model. The latter include various models in Statistical Physics where approximate decoupling of regions creates branching-like effects, or in sparse random graphs where the structure of branching trees is supplemented by additional edges. Finally, various models of evolving populations, which can incorporate geographical or phenotypic structure, competition, cooperation, mutation, reproduction, etc. also give rise to natural models with a rich mathematical structure that incorporate a key branching element.
One possible starting point is a recent study by Evans, Grubel and Wakolbinger to represent the Doob-Martin boundary of one of the simplest discrete tree growth algorithms in terms of continuum tree structures, thereby describing in some sense all conditioned binary tree growth processes. There is a large literature on non-binary tree growth processes, too, including Marchal's tree growth. They relate to multifurcating continuum trees including Duquesne and Le Gall's stable trees. The aim of this example project would be to identify the Doob-Martin boundary of Marchal's tree growth.
Another line of enquiry concerns optimisation problems involving matchings. Certain matching problems (minimal-cost matchings, stable matchings) for the mean-field model of distance have been fruitfully analysed using limiting objects such as Aldous and Steele's "Poisson-weighted infinite tree". It is interesting to ask to what extent this can be extended for example to the "gamma-minimal" or "altruistic" matchings studied recently by Holroyd, Janson and Wästlund. Many more intriguing problems arise when the mean-field model is replaced for example by distances between Poisson points in Euclidean space.
Finally, a third potential project which falls more on the side of the continuous structures concerns the spread of branching particle systems. It is well known that the speed of branching diffusing particle systems is intimately connected to a wide class of reaction-diffusion equations that exhibit propagating fronts. There are many open questions related to the interplay between random particle systems and the analysis of partial differential equations, even in simple geometries.
This project is situated at the Probability side of the general research theme of "Statistics and Applied Probability" with some interaction with the Combinatorics side of "Logic and Combinatorics" and with "Mathematical Analysis."