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

TL;DR

This paper uses renormalization group methods to calculate the decay exponents of reunion probabilities for multiple vicious walkers in dimensions near two, revealing their dependence on the dimension and number of walkers.

Contribution

It provides an $ ext{O}( ext{epsilon}^2)$ calculation of the reunion probability exponents for $p$ vicious walkers in $d=2- ext{epsilon}$ dimensions, connecting them to directed polymer partition functions.

Findings

Calculated $ ext{O}( ext{epsilon}^2)$ exponents for $p$ vicious walkers.

Exact RG result for $p=2$ case, with no $ ext{epsilon}$ expansion.

Determined log corrections at $d=2$.

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.