We control the deposition probability p (the evaporation probability 1 − p) in a restricted solid-on-solid model and monitor the surface width W(L, t) as a function of time t, where L is the system size in d = 4 + 1 dimensions. At p =1/2, the surface becomes flat, following the Edwards and Wilkinson universality class. At p = 0.88, W2(t) grows logarithmically at the beginning and becomes saturated at ln L, showing a scaling $${W^2}\left( {L,t} \right) \sim \ln \left[ {{L^{2\alpha }}f\left( {{t \over {{L^z}}}} \right)} \right]$$ with z ≈ 2.0. For the deposition-only model with p = 1, W2(L,t) shows a power law behavior W2(t) ∼ t2β with a rough interface. With varying p, a smooth-to-rough surface transition is found, implying that d = 4+1 is not the upper critical dimension of the Kardar Parisi Zhang equation.

中文翻译：

---

在 d = 4 + 1 维度上具有各种沉积和蒸发概率的受限固体对固体模型

我们在受限制的固体对固体模型中控制沉积概率 p（蒸发概率 1 - p），并监测表面宽度 W(L, t) 作为时间 t 的函数，其中 L 是系统尺寸，d = 4 + 1 维。在 p = 1/2 处，表面变得平坦，遵循 Edwards 和 Wilkinson 的普遍性类。在 p = 0.88 处，W2(t) 在开始时呈对数增长，并在 ln L 处饱和，显示出缩放 $${W^2}\left( {L,t} \right) \sim \ln \left[ { {L^{2\alpha }}f\left( {{t \over {{L^z}}}} \right)} \right]$$ 与 z ≈ 2.0。对于 p = 1 的仅沉积模型，W2(L,t) 显示幂律行为 W2(t) ∼ t2β，具有粗糙的界面。随着 p 的变化，发现了从光滑到粗糙的表面过渡，这意味着 d = 4+1 不是 Kardar Parisi Zhang 方程的上临界维数。