Scaling of Directed Polymers in Random Media

Directed polymers subject to quenched external impurities (as in a polyelectrolyte in a gel matrix) are examined analytically, and numerically. Transverse fluctuations $\left|\mathrm{x}\right|$ scale with the length $t$ of the polymer as $\left|\mathrm{x}\right|\sim t^{\nu }$. In all dimensions, for sufficiently strong disorder, $\nu$ can be different from the random-walk value of ½. Extensive numerical simulations in two, three, and four dimensions in fact suggest a superuniversal exponent of $\nu =\frac{2}{3}$. The importance of the tree structure of the polymer ensemble is emphasized.