Families of vicious walkers

We consider a generalization of the vicious walker problem in which N random walkers in R d are grouped into p families. Using field-theoretic renormalization group methods we calculate the asymptotic behaviour of the probability that no pairs of walkers from different families have met up to time t . For d > 2, this is constant, but for d < 2 it decays as a power t −α , which we compute to (ε 2 ) in an expansion in ε = 2 − d . The second-order term depends on the ratios of the diffusivities of the different families. In two dimensions, we find a logarithmic decay (ln t ) − and compute exactly.