Schwartz Moshe, Perlsman Ehud
School of Physics and Astronomy, Raymond and Beverly Faculty of Exact Sciences, Tel Aviv University, 69978 Tel Aviv, Israel.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 May;85(5 Pt 1):050103. doi: 10.1103/PhysRevE.85.050103. Epub 2012 May 16.
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.
给出了1 + 4维有向聚合物模型在各种无序类型下的数值结果。这些结果是针对一个比之前考虑的系统尺寸大得多的系统得到的。对于与有向渗流模型相关的极端“强”无序情况（最小 - 最大系统），清晰地揭示了曲折指数ζ = 0.5的期望值，有限尺寸效应非常微弱。对于与Kardar - Parisi - Zhang方程相关的“弱无序”情况，有限尺寸效应更强，但ζ值在0.57附近清晰可见。在强无序系统中，预计系统会从短链的最小 - 最大行为急剧转变为长链的弱无序行为。我们的数值结果与该预期相符。为了完善情况，我们得到了弱无序下的能量涨落指数ω，并且发现ω值在0.14附近。因此，曲折指数和能量涨落指数服从强耦合标度关系2ξ - ω = 1。我们的结果表明，在弱无序情况下1 + 4不是上临界维度，因此4 + 1似乎也不是Kardar - Parisi - Zhang方程的上临界维度。
