Growth of random surfaces

随机表面的生长

• 批准号: 1056390
• 负责人: Alexei Borodin
• 金额: $ 65万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1056390&HistoricalAwards=false
• 关键词: Growth; random; surfaces

The proposal focuses on a class of random growth models of one and two-dimensional interfaces. The growth is local, it has a smoothing mechanism, and the speed of growth may depend on the local slope of the interface.The central role is played by models that enjoy a rich algebraic structure.In many cases this is reflected through determinantal formulas for the correlation functions. Large time asymptotic analysis of such formulas is expected to reveal asymptotic features of the emerging interface in different scales. The main goal of the project is to produce and collect various pieces of information to form a unified picture of the asymptotic behavior of large randomly growing interfaces.The proposal focuses on the study of large time behavior for various models of random interface growth on the plane and in the three-dimensional space. Such models appear naturally in both mathematics and physics, for example when one considers ideal crystal melting, or molecular condensation on a substrate, or traffic evolution. Accurate analysis at large times is typically very difficult, and we concentrate on models with additional algebraic structure that originates in seemingly unrelated parts of mathematics. This structure helps to discover new phenomena that tend to be universal, i.e. present in a much wider range of models. As a result, using sophisticated mathematical tools, we find new universal laws that play the role of the famous bell-shaped curve, and that can often be observed later through physical and numerical experiments.

本文主要研究一类一维和二维界面的随机增长模型。生长是局部的，具有平滑机制，生长速度可能取决于界面的局部斜率。具有丰富代数结构的模型发挥了核心作用。在许多情况下，这是通过相关函数的行列式公式反映出来的。对这些公式的大时间渐近分析有望揭示不同尺度下新界面的渐近特征。该项目的主要目标是产生和收集各种信息，以形成大型随机增长接口的渐近行为的统一图像。本文主要研究平面上和三维空间中各种随机界面生长模型的大时间行为。这样的模型自然地出现在数学和物理学中，例如，当人们考虑理想的晶体熔化，或分子在基质上的凝聚，或交通演化时。通常情况下，准确的分析是非常困难的，我们将重点放在具有附加代数结构的模型上，这些模型起源于看似无关的数学部分。这种结构有助于发现普遍存在的新现象，即存在于更广泛的模型中。结果，使用复杂的数学工具，我们发现了新的普遍规律，这些规律扮演着著名的钟形曲线的角色，这些规律通常可以在以后的物理和数值实验中观察到。