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

摘要

We examine a discrete-time Markovian particle system on ?×?_+ introduced in Defosseux (arXiv: 1012. 0117v1). The boundary {0}×?_+ 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(~∞) (introduced in Borodin and Kuan (Commun. Pure. Appl. Math. 63(7):831-894, 2010, arXiv: 0904. 2607). 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.