Random Motion in Disordered Media: Surface Growth, Ballisticity, and Trapping

该项目发展了统计力学的严格理论，数学模型的研究，通常是随机的，涉及许多微小的粒子或其他物体，理想化和物理上有趣的现象，如冰块融化成水时微观结构的转变。普适性的概念起着基础性的作用：当在空间和时间的适当大尺度上观察这些系统时，某些数学结构从许多不同的随机系统中出现。我们将发展随机生长界面的理论，其中一维表面以随机速率局部生长，同时也受到表面张力等恢复力的影响。在这种情况下，普适性的正确概念由随机偏微分方程Kardar-Parisi-Zhang（KPZ）方程指定。它的后期和大规模行为模拟了许多随机增长接口的通用方面。线系综的构造具有自然吉布斯回复性质，其最低指数曲线是KPZ方程的基本解，这是与Ivan Corwin合作开发的一种技术，将可积系统方法与概率思想结合起来。该提案计划利用这一新的视角来研究KPZ普适类模型中的几种现象，包括布朗最后一次渗流中的老化和去相关，以及艾里片中的聚合物聚结和树状结构。

This project develops the rigorous theory of statistical mechanics, the study of mathematical models, often random in nature and involving many tiny particles or other bodies, that idealize and exemplify physically interesting phenomena such as the transitions in microscopic structure undergone by a cube of ice as it melts into water. The notion of universality plays a fundamental role: certain mathematical structures emerge from many different random systems when these systems are viewed on an appropriately large scale in space and time. We will develop the theory of randomly growing interfaces, where a one-dimensional surface is growing in a local fashion at random rates while also being subject to restoring forces such as surface tension.In this context, the right notion of universality is specified by a stochastic partial differential equation, the Kardar-Parisi-Zhang (KPZ) equation. Its late-time and large-scale behavior models universal aspects of many randomly growing interfaces. The construction of line ensembles having a natural Gibbs resampling property whose lowest indexed curve is the fundamental solution of the KPZ equation is a technique developed in collaboration with Ivan Corwin that marries integrable systems approaches to probabilistic ideas. The proposal plans to exploit this new perspective to investigate several phenomena in models in the KPZ universality class, including aging and decorrelation in Brownian last passage percolation, and polymer coalescence and tree structure in the Airy sheet.