Formation of large-scale random structure by competitive erosion

We study the following one-dimensional model of annihilating particles. Beginning 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, after rescaling by n(1/4) and taking n -> infinity, 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.).