Vicious random walkers in the limit of a large number of walkers

Abstract

The vicious random walker problem on a line is studied in the limit of a large number of walkers. The multidimensional integral representing the probability that the p walkers will survive a time t (denoted Pt(p)) is shown to be analogous to the partition function of a particular one-component Coulomb gas. By assuming the existence of the thermodynamic limit for the Coulomb gas, one can deduce asymptotic formulas for Pt(p) in the large-p, large-t limit. A straightforward analysis gives rigorous asymptotic formulas for the probability that after a time t the walkers are in their initial configuration (this event is termed a reunion). Consequently, asymptotic formulas for the conditional proba..