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Stationary Harmonic Measure and DLA in the Upper Half Plane

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Abstract

In this paper, we introduce the stationary harmonic measure in the upper half plane. By bounding this measure, we are able to define both the discrete and continuous time diffusion limited aggregation (DLA) in the upper half plane with absorbing boundary conditions. We prove that for the continuous model the growth rate is bounded from above by \(o(t^{2+\epsilon })\). Moreover we prove that all the moments are finite for the size of the aggregation. When time is discrete, we also prove a better upper bound of \(o(n^{2/3+\epsilon })\), on the maximum height of the aggregate at time n. An important tool developed in this paper, is an interface growth process, bounding any process growing according to the stationary harmonic measure. Together with [12] one obtains non zero growth rate for any such process.

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Acknowledgements

We would like to thank Itai Benjamini, Noam Berger, Marek Biskup, Rick Durrett, Gady Kozma, Greg Lawler, and Jiayan Ye for fruitful discussions related to this project. We would also like to thank anonymous referee(s) for helpful comments. Research was partially supported by National Science Foundation (Grant Nos. DMS-1407558 and DMS-1812009).

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Authors and Affiliations

  1. Texas A&M University, College Station, TX, USA

    Eviatar B. Procaccia

  2. Peking University, Beijing, China

    Yuan Zhang

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  1. Eviatar B. Procaccia
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  2. Yuan Zhang
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Correspondence to Yuan Zhang.

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Communicated by Alessandro Giuliani.

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E. B. Procaccia: Research supported by NSF grant DMS-1812009. Y. Zhang:The manuscript of this paper was first done when YZ was a visiting assistant professor at Texas A&M University.

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Procaccia, E.B., Zhang, Y. Stationary Harmonic Measure and DLA in the Upper Half Plane. J Stat Phys 176, 946–980 (2019). https://doi.org/10.1007/s10955-019-02327-y

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