Scaling limits of random particle aggregation models

在这个项目中，我们研究二维粒子聚集模型。在这些模型中，我们从单个粒子开始，通常是一个单位圆盘，另一个粒子随机附着在它上面。每次附着之后，都会出现一个新的粒子，并将自己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的典型形状是什么。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，这些模型取决于选定的具体参数。相关的例子包括Hasting-Levitovmodel， Eden model和扩散限制聚集（DLA）。研究这类模型的动机来自于物理科学和生物学，其中分形生长已经在不同的背景下被观察到。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广泛的应用，但缩放限制仍然未知。即使在已知标度极限的情况下，仍然存在一些悬而未决的问题：在这些标度极限附近的渐近波动是什么？这些限制有多普遍，也就是说，它们在多大程度上或多大程度上依赖于特定的微观粒子选择？粒子能随机吗？如果是这样的话，在不改变观测到的大尺度行为的情况下，这些粒子能有多“狂野”？这些模型中有相变吗？什么会导致这样的相变？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。对于一大类参数，我们知道它在宏观上看起来与一个球非常相似，尽管微观上的图像更为复杂。这已经被证明到两个参数和的上界。模拟表明，在这个上限之上，宏观图景发生了巨大变化。这些粒子聚集团似乎只在几个方向上强烈地生长，而不是朝着一个球的方向生长，似乎形成了一棵非常复杂的随机树。为了解决这些问题，本项目的主要方法是使用复杂分析和Schramm-Loewner进化中的loewner链理论方法。洛厄纳链是描述复平面上的紧集生长的一种非常强大和通用的技术。Schramm-Loewner演化（SLE）被引入到统计力学中来描述随机界面。他们已经成功地证明了循环擦除随机游走的缩放极限、均匀生成树中的Peano曲线以及磁化和渗透模型的临界二维模型中的边界界面。施拉姆-洛厄纳演化用洛厄纳链理论来描述是相对容易的。此外，它们具有许多显著的对称性。这两个属性使它们成为应用程序的有趣工具。大多数著名的SLE型曲线是共形不变的，这意味着给定一个域、起点和终点，在该域中从指定的起点到指定的终点只有一个规范的SLE。事实上，这种描述在不同域之间的平滑投影下表现得很好。此外，SLEs是由一个单一的正值唯一参数化的。如果我们选择这个参数为6，那么这些曲线是额外的“局部”，这意味着在触及其域的边界之外，它们的形状在每个域的局部看起来是相同的。我们的目标是利用这些技术和对称性，以获得对二维粒子聚集模型的更深入的理解。

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.