Probability distributions of directed polymers in (1+1)-dimensional random media

In this paper we study directed polymers in (1+1)-dimensional random media with special emphasis on P(x,t), the probability distribution to reach a transverse distance x for a polymer of length t. P(x,t) follows the scaling form P(x,t)x2(t)-1/2f(), where x2(t) is the transverse mean-square displacement, which asymptotically obeys x2(t)t4/3, and is the scaling variable =x/x2(t)1/2. The numerical results indicate that f()exp(-c), where the exponent evolves with such that 2 for0 and 2.5 for1, demonstrating an enhanced Gaussian behavior. We discuss these results in the context of enhanced diffusion.