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 this system and points towards d = 4 as the upper critical dimension of the Kardar–Parisi–Zhang problem. PACS number(s): 05.40+j, 64.60.Ht, 05.70.Ln Typeset using REVTEX 1electronic address: [email protected]