DIRECTED POLYMERS IN A RANDOM MEDIUM - UNIVERSAL SCALING BEHAVIOR OF THE PROBABILITY-DISTRIBUTION

We study the probability distribution P(x, t) for the head of a directed polymer in a random medium in 1 + 1 dimensions. We find that, as a function of the scaling variable u = Absolute value of x/t2/3, the behavior of the probability distribution changes abruptly at u(c) almost-equal-to 1.2. Its universal part is given by P(is-proportional-to) exp(-alphau(y)) where the exponent alpha is distributed for different samples according to P(alpha) = N exp[-f(alpha)u(y)]. For 0.25 less than or similar to u(c), y = 2.1+/-0.1 and the multifractal measure f(alpha) is a convex function with a minimum at alpha(typ), which is the exponent associated with a typical sample of the disorder. For u >> u(c), y = 3.0+/-0.1 and f(alpha) becomes trivial.