Growth of random surfaces

该方案主要研究一维和二维界面的一类随机生长模型。增长是局部的，它具有平滑机制，增长的速度可能取决于界面的局部斜率。中心作用由具有丰富代数结构的模型发挥。在许多情况下，这通过相关函数的行列式公式来反映。对这些公式的大时间渐近分析有望揭示新出现的界面在不同尺度上的渐近特征。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界面渐近行为的统一图景。建议重点研究平面和三维空间中各种随机界面增长模型的大时间行为。这样的模型在数学和物理中都很自然地出现，例如，当人们考虑理想的晶体熔化，或衬底上的分子凝聚，或交通演化时。在很大的时间内进行准确的分析通常是非常困难的，我们将重点放在具有额外代数结构的模型上，这些模型源于数学中看似无关的部分。这种结构有助于发现新的现象，这些现象往往是普遍存在的，即存在于更广泛的模型中。因此，使用复杂的数学工具，我们发现了新的普遍规律，它发挥了著名的钟形曲线的作用，而且通常可以在以后通过物理和数值实验观察到。

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.