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

这个项目发展了严格的统计力学理论，即研究数学模型，这些模型通常是随机的，涉及许多微小的粒子或其他物体，这些模型理想化并举例说明了物理上有趣的现象，例如立方体冰在融化成水时经历的微观结构的转变。普遍性的概念起着基础性的作用：当从空间和时间的适当大尺度来看待许多不同的随机系统时，这些系统就会产生某些数学结构。我们将发展随机增长界面理论，其中一维表面以随机速率以局部方式增长，同时还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一个随机偏微分方程式Kardar-Parisi-Zhang(KPZ)方程来描述。它的后期和大规模行为模拟了许多随机增长的接口的通用方面。构造具有自然Gibbs重采样性质的线系综，其最低索引曲线是KPZ方程的基本解，是与Ivan Corwin合作开发的一种技术，它将可积系统方法与概率思想结合在一起。该提案计划利用这一新的视角来研究KPZ普适性类模型中的几个现象，包括布朗最后一次通过渗流中的老化和去相关，以及艾里表中的聚合物聚合和树状结构。

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.