COLORING OF A ONE-DIMENSIONAL LATTICE BY 2 INDEPENDENT RANDOM WALKERS

A new type of question in random walk theory is formulated and solved for the particular case of a periodic one-dimensional lattice. A ''red '' and a ''blue'' random walker perform simultaneous independent simpl e random walk. Each site is initially uncolored and takes irreversibly the color, red or blue, of the first walker by which it is visited. W e study the resulting coloring of the final state, in which each site is either red or blue, on a ring of L sites, We calculate the probabil ity P(n, L) that site n is red, in the scaling limit L --> infinity wi th n/L fixed, for walkers initially on diametrically opposite sites, W e determine by simulation the number of interfaces (that is, pairs of neighboring red and blue sites), for initial separation a between the walkers. This number is approximate to 2.5 for initially diametrically opposite walkers, and appears to increase logarithmically with L/a.