(2+1)-dimensional growth models in inhomogeneous space

(2 1)维非均匀空间增长模型

• 批准号: 2771439
• 金额: --
• 依托单位: University of Edinburgh
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2771439
• 关键词: 2+1; dimensional; growth; models; inhomogeneous

(2+1)-dimensional (2 space, 1 time dimension) growth and decay models of stepped surfaces in homogeneous space have been studied mathematically for at least a couple of decades. These models can be mapped to certain dynamics on interlacing arrays i.e. an interacting particle system. In the last few years there has been interest in investigating what happens in inhomogeneous space. However our understanding is far from complete, and a general theory is lacking. The aim of this project is to develop some of the required theory. There are several directions:1) Develop a generalisation of the framework of the Schur process [3] and Schur dynamics [4] that will include the models in [1], [2] as special cases. This will involve some symmetric function theory and probability.2) Study scaling limits, both local and global, of the models studied in [1],[2]. This will involve some asymptotic analysis of the correlation kernel obtained therein.3) For discrete-time, in the homogeneous setting, these dynamics are closely related to sampling algorithms of random (uniform) tilings of the Aztec diamond and hexagon for example, see [5]. Develop an analogous theory for tiling models in the inhomogeneous setting i.e. where each tile gets a different weight.

均匀空间中阶跃曲面的（2+1）维（2空间，1时间维）增长和衰减模型已经在数学上研究了几十年。这些模型可以映射到交错阵列上的某些动力学，即相互作用的粒子系统。在过去的几年里，人们对研究非均匀空间中发生的事情很感兴趣。然而，我们的理解是远远不够的，缺乏一个一般性的理论。这个项目的目的是发展一些必要的理论。有几个方向：1）发展舒尔过程[3]和舒尔动力学[4]框架的推广，其中包括[1]，[2]中的模型作为特殊情况。这将涉及到一些对称函数理论和概率。2）研究在[1]，[2]中研究的模型的局部和全局标度极限。这将涉及其中获得的相关核的一些渐近分析。3）对于离散时间，在齐次设置中，这些动态与阿兹特克钻石和六边形的随机（均匀）平铺的采样算法密切相关，例如，参见[5]。为非均匀设置中的平铺模型开发类似的理论，即每个平铺块获得不同的权重。