Scaling limits of random particle aggregation models

在这个项目中，我们研究二维颗粒聚集模型。在这些模型中，我们从一个粒子开始，通常是一个单位圆盘，另一个粒子随机地附着在它上面，每次附着之后，一个新的粒子出现并附着在已经存在的粒子集合上。我们正在研究的主要问题是大颗粒集合体的典型形状。这当然取决于所选择的确切参数。我们正在考虑一系列这样的模型取决于具体选择的参数。相关的例子包括Hasting-Levitov模型，Eden模型和扩散限制聚集（DLA）。研究这类模型的动机来自物理科学和生物学，在不同的背景下观察到分形生长。根据所选择的模型参数，这些模型用于描述例如细菌菌落的生长、电沉积、材料的处置和介电击穿。最后一项是在足够高的电压下，电绝缘体开始充当电导体的效应，尽管在许多有趣的模型中有这样的应用范围，但缩放限制仍然是未知的.即使在这些情况下，标度限制是已知的，仍然有开放的问题：什么是渐近波动围绕这些标度限制？这些极限有多普遍，即它们在多大程度上或多小程度上取决于微观粒子的特定选择？粒子能随机吗？如果是这样的话，这些粒子在不改变大尺度行为的情况下能有多“狂野”？这些模型中是否存在相变？是什么导致了这样的相变？特别是，最后两个问题非常有趣，因为模拟确实表明存在非常有趣的相变。对于一个大类的参数的标度限制是已知的，宏观上看起来非常类似于一个球，虽然微观图片是更复杂的。这已被证明是两个参数之和的上界。模拟表明，超过这个上限，宏观图像发生了显着变化。这些粒子聚集团并不是朝着一个球的方向生长，而是只在几个方向上强烈生长，似乎形成了一棵非常复杂的随机树，本项目主要采用复分析中的Loewner链理论和Schramm-Loewner演化的方法来解决这些问题。Loewner链是描述复平面上增长紧集的一种非常有效和通用的方法，Schramm-Loewner演化（SLE）在统计力学中被用来描述随机界面。他们已经成功地证明了循环擦除随机行走的标度极限，Peano曲线的均匀生成树和临界二维Ising模型的磁化和渗流模型的边界界面。Schramm-Loewner演化相对容易用Loewner链理论来描述。此外，它们还具有许多显著的对称性。这两个性质使它们成为我们应用的有趣工具。最著名的SLE型曲线是共形不变的，这意味着给定一个域，一个起点和一个终点，在这个域中从指定的起点到指定的终点只有一个典型的SLE。事实上，这种描述在不同域之间的光滑双射下表现得很好。此外，SLEs是唯一的参数由一个单一的正值。如果我们选择这个参数为6，那么这些曲线是额外的“本地”，这意味着在击中他们的域的边界之外，他们的形状看起来在每个域的局部相同。我们的目标是使用这些技术和对称性，以获得更深入的了解二维粒子聚集模型。

In this project we investigate two-dimensional particle aggregation models. In these models we startwith a single particle, which is usually a unit disc, to which another particle attaches randomly.After each attachment a new particle appears and attaches itself to the already existing collection ofparticles. The main question we are investigating is what typical shapes large particle collectionswill take. This of course does depend on the exact parameters chosen. We are considering a range ofsuch models depending specific chosen parameters. Relevant examples include the Hasting-Levitovmodel, the Eden model and hopefully diffusion-limiting aggregation (DLA).Motivation to look at these kind of models comes from the physical sciences as well as biologywhere fractal-like growth has been observed in different contexts. Depending on the chosen modelparameters these models are used to describe for instance growth of bacterial colonies,electrodeposition, disposition of materials and the dielectric breakdown. The last item is an effect inwhich under a sufficiently high voltage an electrical insulator starts acting as an electricalconductor.Despite this range of applications in many interesting models scaling limits remain unknown. Evenin those cases where scaling limits are known there are still open questions: What are theasymptotic fluctuation around these scaling limits? How universal are these limits, i.e. how far orhow little do they depend on a specific choice of microscopic particles? Can the particles berandom? If so, how "wild" can these particles be without changing the large scale observedbehaviour? Are there phase transitions in these models? What would cause such a phase transition?Especially, the last two questions are quite interesting, as simulations do suggest the existence of avery interesting phase transitions. For a large class of parameters a scaling limit is know whichmacroscopically looks very similar to a ball, although the microscopic picture is more intricate. Thishas been proven up to an upper bound on a sum of two parameters. Simulations suggest that abovethis upper bound the macroscopic picture changes dramatically. Instead of towards a ball, theseparticle aggregation clusters seems to grow strongly only in a few directions, seemingly resulting ina very complicated random tree.The main approach of this project in order to address these question is to use methods fromLoewner Chain theory from complex analysis and Schramm-Loewner evolutions. Loewner Chainsare a very powerful and general technique to describe growing compact sets in the complex plane.Schramm-Loewner evolutions (SLE) have been introduced in statistical mechanics to describerandom interfaces. They have successfully proven the be the scaling limits of loop erased randomwalks, Peano curves the in the uniform spanning tree and boundary interfaces in the critical twodimensionalIsing model for magnetisation and percolation models. Schramm-Loewner evolutionsare relatively easy to describe using the theory of Loewner Chain. Moreover, they possess manyremarkable symmetries. Both of these properties render them interesting tools to our application.Most famously SLE-type curves are conformally invariant, which means that given a domain, astarting point and an end point there is only one canonical SLE in this domain from the specifiedstarting to the specified end point. In fact, this description behaves nicely under smooth bijectionsbetween different domains. Furthermore, SLEs are uniquely parametrised by a single positive value.If we choose this parameter to be six, then these curves are additionally "local" which means thatoutside of hitting the boundary of their domain their shape looks locally the same in every domain.Our goal is to use these techniques and symmetries in order to obtain a deeper understanding oftwo-dimensional particle aggregation models.