Some contributions at the interface of combinatorial probability and continuum structures

概率论中离散结构和连续结构之间的相互作用有着悠久的历史，可以追溯到世纪的中心极限定理的早期版本，作为一种通过连续近似来描述大型离散结构的方法。在整个世纪，这个概念有许多扩展和抽象，包括布朗运动作为随机游动的极限，扩散近似人口模型，连续表示的大随机图和连续模型的平均场或格子模型的统计物理。这样的“标度限制”往往是普遍的意义上，他们捕捉大系统的行为大类离散模型，不同的地方behavior.Of本项目的特别相关的分支结构，特别是基本的系谱树，以及结构中出现的分支作为一种技术，而不是模型的内在特征的区域的最新发展。后者包括统计物理学中的各种模型，其中区域的近似解耦产生了类似分支的效果，或者在稀疏随机图中，分支树的结构由额外的边缘补充。最后，进化种群的各种模型，可以包括地理或表型结构，竞争，合作，突变，繁殖等，也会产生具有丰富数学结构的自然模型，其中包含关键分支元素。一个可能的出发点是埃文斯，格鲁贝尔和Wakolbinger最近的一项研究，以连续树结构表示最简单的离散树生长算法之一的Doob-Martin边界，从而在某种意义上描述了所有条件二叉树的生长过程。有大量的文献对非二叉树的增长过程，太，包括马夏尔的树的增长。它们涉及到多叉连续树，包括迪克讷和勒加尔的稳定树。本示例项目的目的是确定马夏尔树生长的Doob-Martin边界。另一条调查线涉及匹配的优化问题。某些匹配问题（最小成本匹配，稳定匹配）的平均场模型的距离已卓有成效地分析使用限制对象，如奥尔德斯和斯蒂尔的“泊松加权无限树”。有趣的是，这在多大程度上可以扩展到例如Holroyd，Janson和Wästlund最近研究的“gamma-minimal”或“altruistic”匹配。当平均场模型被欧几里德空间中泊松点之间的距离所取代时，会出现许多更有趣的问题。最后，福尔斯更倾向于连续结构的第三个潜在项目涉及分支粒子系统的扩展。众所周知，分支扩散粒子系统的速度与一类具有传播前沿的反应扩散方程密切相关。有许多悬而未决的问题与随机粒子系统之间的相互作用和偏微分方程的分析，即使在简单的几何形状。这个项目是位于概率方面的一般研究主题的“统计和应用概率”与一些互动的组合数学方面的“逻辑和组合数学”和“数学分析。"

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."