Reunion of Vicious Walkers: Results from $ε$-Expansion -

Abstract

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