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 $|\mathrm{x}|$ scale with the length $t$ of the polymer as $|\mathrm{x}|\ensuremath{\sim}{t}^{\ensuremath{\nu}}$. In all dimensions, for sufficiently strong disorder, $\ensuremath{\nu}$ can be different from the random-walk value of \textonehalf{}. Extensive numerical simulations in two, three, and four dimensions in fact suggest a superuniversal exponent of $\ensuremath{\nu}=\frac{2}{3}$. The importance of the tree structure of the polymer ensemble is emphasized.