OVERLAP, DISORDER, AND DIRECTED POLYMERS - A RENORMALIZATION-GROUP APPROACH

The overlap of a (d+1)-dimensionaI directed polymer of length t in a r andom medium is studied using a renormalization group approach. In d>2 it vanishes at T-c for t-->infinity as t(Sigma) where Sigma=[(d-1)/(3 -2d)](d/z) and z is the transverse spatial rescaling exponent. The sam e formula holds in d=1 for any finite temperature and it agrees with p revious numerical simulations at d=1. This value of Sigma for d>2 is u p to one loop but an exact scaling relation is obtained. We also obtai n the scaling exponent for mutual repulsion of two chains in the rando m medium and the behavior of overlap around T-c.