Random Growth Models

随机增长模型

• 批准号: RGPIN-2022-03633
• 负责人: Dauvergne, Duncan
• 金额: $ 2.33万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758283
• 关键词: Random; Growth; Models

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.

随机增长是概率和数学物理领域最活跃的研究领域之一。研究的模型基于各种物理现象，例如细菌菌落生长、感染传播和化学反应。过去二十五年来，该领域的研究不断取得突破，揭示了与许多其他领域的联系，包括随机微分方程、代数组合学和表示论。这些突破中的第一个突破使用了少数一维界面增长模型中存在的精确可解结构来找到高度分布的限制公式。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联系。在这些初步突破之后，研究人员不断完善对这些模型的确切可解结构的理解，以获得对极限图像更丰富、更深入的理解。虽然通过研究精确可解模型可以学到很多东西，但大多数随机增长模型不属于这一类。此外，许多不可解模型揭示了精确可解模型中不存在的现象。近几十年来，对这些模型的研究也一直在稳步进行，其中以形状定理、标度关系和测地线结构的新证明为重点。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。来自不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显着结果，例如解决 tasep 的慢键问题，以及艾里线系综的布朗吉布斯性质的存在。在这方面，奥特曼、维拉格和我混合使用了可解和不可解的技术来构建有向景观，这是一维增长模型类中最丰富的限制对象。有向景观包含了所有以前理解的作为边际的限制分布，它的构建开辟了许多新的、令人兴奋的研究途径。拟议的研究计划继续将可解模型和不可解模型的研究思想相结合。该计划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。从研究精确可解模型中获得的见解对于研究不可解模型很有用，反之亦然。具体问题包括定向景观中测地线的分类、通过纯概率标准描述定向景观的特征，以及理解感染传播模型中的极限形状和扩散率的影响。