Competitive erosion models a random interface sustained in equilibrium by equal and opposite
pressures on each side of the interface. Here we study the following one dimensional
version. Begin with all sites of 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 has a certain fractal nature which after scaling by n^1/4 and taking a
limit, has an explicit description in terms of alternating extrema of Brownian motions.
This is joint work with Shirshendu Ganguly and Lionel Levine.