Asymptotics of a Discrete-Time Particle System Near a Reflecting Boundary

We examine a discrete-time Markovian particle system on N × Z+ in-troduced in [8]. The boundary {0} × Z+ acts as a reflecting wall. The particle system lies in the Anisotropic Kardar-Parisi-Zhang with a wall universality class. After projecting to a single horizontal level, we take the long–time asymptotics and obtain the discrete Jacobi and symmetric Pearcey kernels. This is achieved by showing that the particle system is identical to a Markov chain arising from representations of O(∞) (intro-duced in [6]). The fixed–time marginals of this Markov chain are known to be determinantal point processes, allowing us to take the limit of the correlation kernel. We also give a simple example which shows that in the multi-level case, the particle system and the Markov chain evolve differently.