Scaling limits of random and deterministic diffusion processes on the integer lattice

首席研究员将研究固体材料中微观相互作用的数学模型。更具体地说，PI将研究此类模型的比例限制，这些模型捕捉到微观结构的宏观效应。PI对统计物理产生的三类重要模型感兴趣。首先，有金属腐蚀和晶体形成的团簇生长模型。第二，随机组合材料的电学性质模型。第三，有一些模型位于前两个类别的交叉点上，比如那些关于多孔介质中流体流动的模型。围绕着这些模型的许多悬而未决的问题现在引起了概率论的极大兴趣。PI的目的是通过(1)使用计算机模拟来识别新现象和(2)应用分析、组合学和概率学的工具来严格理解新现象来解决其中的一些问题。上述模型有一个共同的主题；每个模型都是整数格上的扩散过程，其标度极限可以解释为非线性椭圆型偏微分方程(PDE)的解。PI主要对三种模型感兴趣：阿贝尔沙堆、随机环境中的随机游动和首次通过渗流。PI将研究扩散过程的渐近统计量与极限偏微分方程解的正则性之间的相互作用。由于这里的双方紧密相连，它们之间的成果转移往往是可能的和富有成效的。策略将是仔细调整限制偏微分方程的规律性机制以适应离散的环境，因为这偶尔会导致新现象的发现。

The principal investigator will study mathematical models of microscopic interactions in solid materials. More specifically, the PI will study the scaling limits of such models, which capture macroscopic effects of microscopic structure. The PI is interested in three important classes of models arising from statistical physics. First, there are cluster growth models for corrosion of metals and crystal formation. Second, there are models of electrical properties of randomly composited materials. Third, there are models which lie in the intersection of the previous two categories, such as those for fluid flow in porous media. Many unresolved questions surround these models and are now of great interest in probability theory. The PI aims to resolve some of these questions by (1) using computer simulation to identify new phenomena and (2) applying tools from analysis, combinatorics, and probability to rigorously understand the new phenomena.The models alluded to above share a common theme; each is a diffusion processes on the integer lattice whose scaling limit can be interpreted as the solution of a nonlinear elliptic partial differential equation (PDE). The PI is primarily interested in three models: the Abelian sandpile, random walks in random environments, and first passage percolation. The PI will study the interplay between the asymptotic statistics of the diffusion process and the regularity properties of the solutions of the limiting PDE. Since the two sides here are tightly coupled, it is often possible and fruitful to transfer results between them. The strategy will be to carefully adapt the regularity machinery of the limiting PDE to the discrete setting, since this occasionally leads to the discovery of new phenomena.