Stationary DLA is Well Defined
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Stationary DLA is Well Defined Eviatar B. Procaccia1,2 · Jiayan Ye2 · Yuan Zhang3 Received: 23 January 2020 / Accepted: 24 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we construct an infinite stationary diffusion limited aggregation (SDLA) on the upper half planar lattice, growing from an infinite line, with local growth rate proportional to the stationary harmonic measure. This model was suggested by Itai Benjamini. The main issue is a known problem in DLA models, the long range effects of large arms. In this paper we overcome this difficulty via a multi-scale argument controlling the dynamical discrepancies created on all scales while running two coupled SDLA on different starting configurations. Keywords Diffusion limited aggregation · Stationary harmonic measure · Interacting particle system
1 Introduction Diffusion limited aggregation (DLA) is a set-valued process first defined by Witten and Sander [12] in order to study physical systems where the growth are governed by diffusion. DLA is defined recursively as a process on subsets of Z2 . Starting from A0 = {(0, 0)}, at each time a new point an+1 sampled from the harmonic probability measure on the outer
Communicated by Ivan Corwin. Research supported by NSF Grant DMS-1812009 and NSFC Young Scientists Fund (Grant Number 11901012).
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Yuan Zhang [email protected] Eviatar B. Procaccia [email protected] http://www.math.tamu.edu/~procaccia Jiayan Ye [email protected] http://www.math.tamu.edu/~tomye
1
Technion - Israel Institute of Technology, Haifa, Israel
2
Texas A&M University, College Station, TX, USA
3
Peking University, Beijing, China
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vertex boundary of An is added to An . Intuitively, an+1 is the first place that a random walk starting from infinity visits ∂ out An . In many experiments and real world phenomena the aggregation grows from some initial boundary instead of a single point, e.g. ions diffusing in liquid until they connect a charged container floor (see [2] for numerous examples). Different aggregation processes, such as Eden and Internal DLA, with boundaries were studied in [1,3], and universal phenomena such as a.s. non existence of infinite trees were proved. Motivated by a question from Itai Benjamini (through private communication), in this paper we construct an infinite stationary DLA (SDLA) on the upper half planar lattice, growing from an infinite line. Along the way we prove that this infinite stationary DLA can be seen as a limit of DLA in the upper half plane growing from a long finite line. This allows one to use the more symmetric and amenable model of SDLA to study local behavior of DLA. In addition SDLA admits new phenomena not observed in the full lattice DLA. One such interesting conjectured phenomenon, which results from the competition between different trees in the SDLA, is that eventually (and in finite time) every tree in the SDLA ceases to grow. The main difficulties we encountered in this paper are also known
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