The Annals of Probability

Formation of large-scale random structure by competitive erosion

Shirshendu Ganguly, Lionel Levine, and Sourav Sarkar

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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.).

Article information

Source
Ann. Probab., Volume 47, Number 6 (2019), 3649-3704.

Dates
Received: April 2018
First available in Project Euclid: 2 December 2019

Permanent link to this document
https://projecteuclid.org/euclid.aop/1575277339

Digital Object Identifier
doi:10.1214/19-AOP1342

Mathematical Reviews number (MathSciNet)
MR4038040

Subjects
Primary: 60G50: Sums of independent random variables; random walks 60J65: Brownian motion [See also 58J65] 82C22: Interacting particle systems [See also 60K35] 82C24: Interface problems; diffusion-limited aggregation 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

Keywords
Alternating extrema annihilating particle system Brownian motion competitive erosion diffusion-limited aggregation macroscopic interface

Citation

Ganguly, Shirshendu; Levine, Lionel; Sarkar, Sourav. Formation of large-scale random structure by competitive erosion. Ann. Probab. 47 (2019), no. 6, 3649--3704. doi:10.1214/19-AOP1342. https://projecteuclid.org/euclid.aop/1575277339


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