Tanner Reese: Crystal Growth on Locally Finite Partially Ordered Sets

One can model a crystal growing in a corner using a set-valued Markov process. By analyzing the asymptotic behavior of this growth process, one can come to understand its macroscopic properties. This model has close connections to other statistical models such as Totally Asymmetric Simple Exclusion Processes (TASEP), Last-Passage Percolation (LPP), and random matrices. We will investigate how one can generalize this model to certain partially ordered monoids, demonstrate the existence of a shape function, and prove bounds for the mean and fluctuations of the stopping times.