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November 2019 Formation of large-scale random structure by competitive erosion
Shirshendu Ganguly, Lionel Levine, Sourav Sarkar
Ann. Probab. 47(6): 3649-3704 (November 2019). DOI: 10.1214/19-AOP1342

Abstract

We study the following one-dimensional model of annihilating particles. Beginning with all sites of $\mathbb{Z}$ uncolored, a blue particle performs simple random walk from $0$ until it reaches a nonzero red or uncolored site, and turns that site blue; then a red particle performs simple random walk from $0$ until it reaches a nonzero blue or uncolored site, and turns that site red. We prove that after $n$ blue and $n$ red particles alternately perform such walks, the total number of colored sites is of order $n^{1/4}$. The resulting random color configuration, after rescaling by $n^{1/4}$ and taking $n\to \infty $, has an explicit description in terms of alternating extrema of Brownian motion (the global maximum on a certain interval, the global minimum attained after that maximum, etc.).

Citation

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Shirshendu Ganguly. Lionel Levine. Sourav Sarkar. "Formation of large-scale random structure by competitive erosion." Ann. Probab. 47 (6) 3649 - 3704, November 2019. https://doi.org/10.1214/19-AOP1342

Information

Received: 1 April 2018; Published: November 2019
First available in Project Euclid: 2 December 2019

zbMATH: 07212169
MathSciNet: MR4038040
Digital Object Identifier: 10.1214/19-AOP1342

Subjects:
Primary: 60G50, 60J65, 82C22, 82C24, 82C41

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 6 • November 2019
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