Reunion of Vicious Walkers: Results from $\epsilon$-Expansion -

Abstract

The anomalous exponent, $\eta_{p}$, for the decay of the reunion probability of $p$ vicious walkers, each of length $N$, in $d$ $(=2-\epsilon)$ dimensions, is shown to come from the multiplicative renormalization constant of a $p$ directed polymer partition function. Using renormalization group(RG) we evaluate $\eta_{p}$ to $O(\epsilon^2)$. The survival probability exponent is $\eta_{p}/2$. For $p=2$, our RG is exact and $\eta_p$ stops at $O(\epsilon)$. For $d=2$, the log corrections are also determined. The number of walkers that are sure to reunite is 2 and has no $\epsilon$ expansion.