Families of Vicious Walkers

TL;DR

This paper analyzes the asymptotic probability that multiple families of random walkers in various dimensions do not meet, using field-theoretic methods to compute decay exponents and their dependence on diffusivities.

Contribution

It extends the vicious walker problem to multiple families and calculates decay exponents in different dimensions using renormalisation group techniques.

Findings

For d>2, the probability remains constant over time.

For d<2, the probability decays as a power law with exponent alpha.

In two dimensions, the decay is logarithmic, with an exactly computed exponent alpha'.

Abstract

We consider a generalisation of the vicious walker problem in which N random walkers in R^d 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^(-alpha), which we compute to O(epsilon^2) in an expansion in epsilon=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)^(-alpha'), and compute alpha' exactly.