Random Growth Models

点击翻译按钮获取中文摘要

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.