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.).
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
