Overlap, Disorder and Directed Polymers: A Renormalization Group Approach -

The overlap of a $d+1$ dimensional directed polymer of length $t$ in a random medium is studied using a Renormalization Group approach. In $d>2$ it vanishes at $T_c$ for $t\rightarrow \infty$ as $t^{\Sigma}$ where $\Sigma=[\frac{d-1}{3-2d}]\frac{d}{z}$ and $z$ is the transverse spatial rescaling exponent. The same formula holds in $d=1$ for any finite temperature and it agrees with previous numerical simulations at $d=1$. Among other results we also determine the scaling exponent for mutual repulsion of two chains in the random medium.