Families of Vicious Walkers

We consider a generalisation of the vicious walker problem in which N random walkers in R are grouped into p families. Using field-theoretic renormalisation 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, which we compute to O(ε) 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 ᾱ exactly.