Directed polymers in high dimensions

We study directed polymers subject to a quenched random potential in d transversal dimensions. This system is closely related to the Kardar-Parisi-Zhang equation of nonlinear stochastic growth. By a careful analysis of the perturbation theory we show that physical quantities develop singular behavior for d{→}4. For example, the universal finite-size amplitude of the free energy at the roughening transition is proportional to {}4-d. This shows that the dimension d=4 plays a special role for the Kardar-Parisi-Zhang problem. {} 1996 The American Physical Society.