Universality and crossover of directed polymers and growing surfaces

We study Kardar-Parisi-Zhang surfaces on Euclidean lattices and directed po lymers on hierarchical lattices subject to different distributions of disor der, showing that universality holds, at odds with recent results on Euclid ean lattices. Moreover, we find the presence of a slow (power-law,) crossov er toward the universal values of the exponents and verify that the exponen t governing such crossover is universal too. In the limit of a (1 + epsilon )-dimensional system we find, both numerically and analytically, that the c rossover exponent is 1/2. [S0031-9007(99)09043-2].