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 $\mathbb{Z}$ 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 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 a joint work with Shirshendu Ganguly and Lionel Levine.