Random Growth Models

随机增长是概率论和数学物理中最活跃的研究领域之一。所研究的模型是基于各种物理现象，如细菌菌落生长，感染传播和化学反应。在过去的25年里，该领域的研究取得了多次突破，揭示了与许多其他领域的联系，包括随机微分方程、代数组合学和表示理论。这些突破中的第一个是使用少量一维界面生长模型中存在的精确可解结构来找到高度分布的极限公式。令人惊讶的是，发现的分布与随机矩阵理论中已经看到的分布相匹配，尽管这两个世界之间几乎没有表面上的联系。在这些最初的突破之后，研究人员逐渐完善了他们对这些模型的精确可解结构的理解，从而对极限图景有了更丰富、更深入的理解。虽然我们可以从研究精确可解模型中学到很多东西，但大多数随机增长模型并不属于这一类。此外，许多不可解模型揭示了在完全可解模型中不存在的现象。近几十年来，对这些模型也有稳定的研究，突出的是形状定理，缩放关系和测地线结构的新证明。虽然最初研究这两种模型的小组之间没有太多重叠，但在过去十年中，这种情况开始发生变化。来自非可解模型研究的概率和几何思想已经开始与关键的可解输入结合使用，以产生关于可解模型的新的和显著的结果，例如tasep的慢键问题的解决，以及Airy线系综的布朗吉布斯性质的存在。在这种情况下，Ortmann， Virag和我混合使用了可解和不可解的技术来构建定向景观，这是一维增长模型中最丰富的限制对象。定向景观包含了所有以前理解的边缘限制分布，它的构建开辟了许多新的和令人兴奋的研究途径。拟议的研究计划延续了这条路线，将研究可解模型和不可解模型的思想结合起来。该项目将研究这两个群体的模型，重点是发现和理解新的和以前无法理解的现象。从研究完全可解模型中获得的洞察力对于研究不可解模型是有用的，反之亦然。具体问题包括定向景观中测地线的分类，通过纯概率标准表征定向景观，以及理解感染传播模型中的极限形状和扩散速率的影响。

Random growth is one of the most active research areas within probability and mathematical physics. The models studied are based on a variety of a physical phenomena such as bacterial colony growth, infection spread, and chemical reactions. The past twenty-five years of research in the area has been marked by repeated breakthroughs, revealing connections with many other fields, including stochastic differential equations, algebraic combinatorics, and representation theory. The first of these breakthroughs used exactly solvable structure present in a small handful of one-dimensional interface growth models to find limiting formulas for height distributions. Surprisingly, the distributions found matched with distributions already seen in random matrix theory, despite there being almost no superficial connection between the two worlds. After these initial breakthroughs, researchers steadily refined their understanding of the exactly solvable structure of these models to get a richer and deeper understanding of the limiting picture. While there is a great deal to be learned from studying exactly solvable models, most random growth models do not fall into this category. Moreover, many non-solvable models reveal phenomena that are not present in exactly solvable models. In recent decades, there has also been steady research on these models, highlighted by new proofs of shape theorems, scaling relationships, and geodesic structures. While initially there was not much overlap between groups studying these two types of models, over the past ten years, this has started to change. Probabilistic and geometric ideas from the study of non-solvable models have started to be used in conjunction with key solvable inputs to produce new and remarkable results about solvable models, such as the resolution of the slow bond problem for tasep, and the existence of the Brownian Gibbs property for the Airy line ensemble. In this vein, Ortmann, Virag, and I used a mixture of solvable and non-solvable techniques to construct the directed landscape, the richest limiting object in the class of one-dimensional growth models. The directed landscape contains all previously understood limiting distributions as marginals, and its construction opens up many new and exciting avenues of research. The proposed research program continues this line of combining ideas from the study of solvable and non-solvable models. The program will study models from both these groups, with a focus on finding and understanding new and previously inaccessible phenomena. Insight gained from studying exactly solvable models is useful for studying non-solvable models, and vice versa. Specific problems include the classification of geodesics in the directed landscape, characterizing the directed landscape via purely probabilistic criteria, and understanding limit shapes and the effects of diffusion rate in infection spread models.