Uninfected random walkers in one dimension

We consider a system of unbiased diffusing walkers (A0 <-> 0A) in one dimension with random initial conditions. We investigate numerically the relation between the fraction of walkers, U(t), which have never encountered another walker up to time t, calling such walkers ``uninfected'' and the fraction of sites, P(t), which have never been visited by a diffusing particle. We extend our study to include the A + B -> 0 diffusion-limited reaction in one-dimension, with equal initial densities of A and B particles distributed homogeneously at t=0. We find U(t) \simeq [P(t)]^\gamma, with \gamma \simeq 1.39, in both models, though there is evidence that a smaller value of \gamma is required for t -> infinity.