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

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Abstract

We examine a discrete-time Markovian particle system on ℕ×ℤ+ introduced in Defosseux (arXiv:1012.0117v1). 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(∞) (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.

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Notes

  1. ℕ denotes the non-negative integers and ℤ+ denotes the positive integers.

  2. Strictly speaking, Proposition 3.11 proves (2) without showing that T ϕ has non-negative entries. A better term would be “signed Markov chain,” but this is not standard terminology. In any case, Proposition 3.1 below will show that for the ϕ studied in this paper, T ϕ is a bona-fide Markov chain.

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Acknowledgements

The author would like to thank Alexei Borodin, Manon Defosseux, Ivan Corwin and the referees for helpful comments.

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  1. Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA, 02138, USA

    Jeffrey Kuan

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  1. Jeffrey Kuan
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Correspondence to Jeffrey Kuan.

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Kuan, J. Asymptotics of a Discrete-Time Particle System Near a Reflecting Boundary. J Stat Phys 150, 398–411 (2013). https://doi.org/10.1007/s10955-012-0681-9

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