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({ξ}), where 〈x²(t)〉 is the transverse mean-square displacement, which asymptotically obeys 〈x²(t)〉{~}t4/3, and {ξ} is the scaling variable {ξ}=x/〈x²(t)〉1/2. The numerical results indicate that f({ξ}){}exp(-cξ^δ), where the exponent {δ} evolves with {ξ} such that {δ}{}2 for {ξ}{→}0 and {δ}{}2.5 for {ξ}{}1, demonstrating an ``enhanced'' Gaussian behavior. We discuss these results in the context of enhanced diffusion.