Formation of large-scale random structure by competitive erosion

TL;DR

This paper analyzes a one-dimensional competitive erosion model where particles of two colors perform random walks and change site colors upon contact, revealing a large-scale structure related to Brownian motion extrema.

Contribution

It introduces a detailed analysis of the large-scale structure formed by competitive erosion, linking it to extrema of Brownian motion, and provides explicit asymptotic descriptions.

Findings

Number of colored sites grows as n^{1/4}

Rescaled configuration converges to a process described by Brownian extrema

Explicit connection between particle dynamics and Brownian motion extrema

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.).