Universality and Crossover of Directed Polymers and Growing Surfaces

We study Kardar-Parisi-Zhang surfaces on Euclidean lattices and directed polymers on hierarchical lattices subject to different distributions of disorder, showing that universality holds, at odds with recent results on Euclidean lattices. Moreover, we find the presence of a slow (power-law) crossover toward the universal values of the exponents and verify that the exponent governing such crossover is universal too. In the limit of a ( 1+ε)-dimensional system we find, both numerically and analytically, that the crossover exponent is $1/2$.