Vicious walks with long-range interactions

Abstract

The asymptotic behaviour of the survival or reunion probability of vicious walks with short-range interactions is generally well studied. In many realistic processes, however, walks interact with a long ranged potential that decays in $d$ dimensions with distance $r$ as $r^{-d-\sigma}$. We employ methods of renormalized field theory to study the effect of such long range interactions. We calculate, for the first time, the exponents describing the decay of the survival probability for all values of parameters $\sigma$ and $d$ to first order in the double expansion in $\epsilon=2-d$ and $\delta=2-d-\sigma$. We show that there are several regions in the $\sigma-d$ plane corresponding to different scalings for survival and reunion probabilities. Furthermore, we calculate the leading logarithmic corrections for the first time.