Upper critical dimension of the Kardar-Parisi-Zhang equation

Numerical results for the directed polymer model in 1+4 dimensions in various types of disorder are presented. The results are obtained for a system size that is considerably larger than considered previously. For the extreme "strong" disorder case (min-max system), associated with the directed percolation model, the expected value of the meandering exponent, ζ=0.5, is clearly revealed, with very weak finite size effects. For the "weak disorder" case, associated with the Kardar-Parisi-Zhang equation, finite size effects are stronger, but the value of ζ is clearly seen in the vicinity of 0.57. In systems with strong disorder it is expected that the system will cross over sharply from min-max behavior at short chains to weak disorder behavior at long chains. Our numerical results agree with that expectation. To complete the picture we obtain the energy fluctuation exponent ω for weak disorder, and we find that the value of ω is in the vicinity of 0.14. Thus, the meandering exponent and the energy fluctuation exponent obey the strong coupling scaling relation 2ξ-ω=1. Our results indicate that 1+4 is not the upper critical dimension in the weak disorder case, and thus 4+1 does not seem to be the upper critical dimension for the Kardar-Parisi-Zhang equation.