Some contributions at the interface of combinatorial probability and continuum structures

离散结构和连续结构在概率上的相互作用有着悠久的历史，最早可以追溯到18世纪和19世纪的中央极限定理的早期版本，该定理是一种用连续统近似来描述大型离散结构的方法。在20世纪的过程中，这一概念有了许多扩展和抽象，包括作为随机行走极限的布朗运动，人口模型的扩散近似，大型随机图的连续统表示，以及统计物理平均场或格子模型中的连续统模型。这种“比例限制”通常是普遍的，因为它们捕捉了大类离散模型的大系统行为，这些模型的局部行为不同。与本项目特别相关的是分支结构领域的最新发展，特别是对于潜在的谱系树，以及其中分支作为一种技术而不是模型的内在特征出现的结构。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类似分支的效果，或者在稀疏随机图中，分支树的结构由附加的边补充。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，也产生了具有丰富数学结构的自然模型，其中包含了关键的分支元素。一个可能的起点是Evans，Grubel和Wakolbinger最近的一项研究，该研究用连续统树结构来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有条件的二叉树生长过程。也有大量关于非二叉树生长过程的文献，包括Marchal的树生长。它们涉及多分支连续统树，包括Duqune和Le Gall的稳定树。这个示例项目的目的将是确定Marchal的树木生长的Doob-Martin边界。另一条询问路线与涉及匹配的优化问题有关。利用Aldous和Steele的“泊松加权无限树”等极限对象，对距离平均场模型的某些匹配问题(最小代价匹配、稳定匹配)进行了卓有成效的分析。有趣的是，这可以扩展到什么程度，例如，Holroyd，Janson和Wästlund最近研究的“伽马极小”或“利他”匹配。当平均场模型被替换为欧几里得空间中泊松点之间的距离时，出现了更多有趣的问题。最后，第三个可能的项目更多地落在连续结构的一侧，涉及分支粒子系统的传播。众所周知，分枝扩散粒子系统的速度与一大类呈现传播前沿的反应扩散方程密切相关。关于随机粒子系统之间的相互作用和偏微分方程的分析，有许多悬而未决的问题，即使在简单的几何中也是如此。这个项目位于一般研究主题“统计学和应用概率”的概率方面，与“逻辑和组合数学”的组合方面和“数学分析”有一些互动。

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."