Uninfected random walkers in one dimension

We consider a system of unbiased diffusing walkers (A(Phi)<-->(Phi)A) 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--> Phi diffusion-limited reaction in one dimension, with equal initial densities of A and B particles distributed homogeneously at t=0. We find U(t) approximately [P(t)]gamma, with gamma approximately 1.39, in both models, though there is evidence that a smaller value of gamma is required for t-->infinity.