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Stationary Harmonic Measure and DLA in the Upper Half Plane

  • Eviatar B. Procaccia
  • Yuan ZhangEmail author
Article
  • 33 Downloads

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.

Notes

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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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Texas A&M UniversityCollege StationUSA
  2. 2.Peking UniversityBeijingChina

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