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

均匀空间中阶梯表面的(2+1)维(2空间，1时间维)增长和衰减模型已经被数学研究了至少几十年。这些模型可以映射到交错阵列上的特定动力学，即相互作用的粒子系统。在过去的几年里，人们一直对研究非均匀空间中发生的事情感兴趣。然而，我们的认识还远远不完整，缺乏一个普遍的理论。这个项目的目的是发展一些必要的理论。有几个方向：1)推广舒尔过程[3]和舒尔动力学[4]的框架，将[1]、[2]中的模型作为特例包括在内。这将涉及到一些对称函数理论和概率。2)研究[1]、[2]中所研究模型的局部和全局标度极限。3)对于离散时间，在均匀设置下，这些动力学与随机(均匀)平铺的采样算法密切相关，例如阿兹特克钻石和六边形，见[5]。为非均匀环境中的瓷砖模型开发一个类似的理论，即每个瓷砖获得不同的权重。

(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.