Phase transition of directed polymer in random potentials on 4+1 dimensions

Finite-temperature-directed polymer in random potentials is described by a transfer matrix method. On 4+1 dimensions, the evidence for a finite-temperature phase transition is found at Tc≈0.18, where the free energy fluctuation grows logarithmically as a function of time t. When T⪡Tc, the fluctuation of the free energy grows as tω with ω≈0.156. The phase transition of the restricted solid-on-solid model, which is closely related to the directed polymer problem through the Kardar–Parisi–Zhang equation, is also discussed.