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 Tc for t{→}{∞} as t^Σ where {Σ}=[(d-1)/(3-2d)](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. This value of {Σ} for d{}2 is up to one loop but an exact scaling relation is obtained. We also obtain the scaling exponent for mutual repulsion of two chains in the random medium and the behavior of overlap around Tc.