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

(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.