Universality of fluctuations in the Kardar-Parisi-Zhang class in high dimensions and its upper critical dimension

We show that the theoretical machinery developed for the Kardar-Parisi-Zhang (KPZ) class in low dimensions is obeyed by the restricted solid-on-solid model for substrates with dimensions up to $d=6$. Analyzing different restriction conditions, we show that the height distributions of the interface are universal for all investigated dimensions. It means that fluctuations are not negligible and, consequently, the system is still below the upper critical dimension at $d=6$. The extrapolation of the data to dimensions $d\ge 7$ predicts that the upper critical dimension of the KPZ class is infinite.

Figure 1

(a) Effective growth exponent defined as $dlnW/dlnt$ for the RSOS model in $d+1$ dimensions with a height restriction $m=4$. The dashed lines are the estimates for $\beta$. (b) Curves used to determine the asymptotic interface velocity. The solid lines are linear regressions. (c) Determination of the quantity ${\Gamma }^{\beta }\langle \chi \rangle$. The horizontal lines are the extrapolated values. (d) Scaling of the shift in relation to the asymptotic distribution. The dashed lines are power laws $t^{-\beta }$ using $\beta$ exponents given in Table 1.

Figure 2

Dimensionless cumulant ratios (a) $R={\langle h^2\rangle }_c/{\langle h\rangle }^2$, (b) $\left|S\right|=|\langle h^3\rangle |/{\langle h^2\rangle }_c^{3/2}$, and (c) $K={\langle h^4\rangle }_c/{\langle h^2\rangle }_c^2$ against the inverse of time for the RSOS model in different dimensions, indicated near the respective curves in (a). Data for height restriction parameters of Table 1 are shown for the respective dimension, with the smaller values represented by the dashed lines. (d) Cumulant ratios as functions of the substrate dimension. Data for $d=1$ and 2 were taken from Refs. [9] and [15], respectively.

Figure 3

Determination of nonuniversal parameters in the RSOS model with a height restriction $m=2$ for three- and four-dimensional substrates. The lines are parabolic (left) or linear (right) regressions to determine the parameters $\lambda$ and $A$, respectively.

Figure 4

Determination of (a) first and (b) second cumulants of $\chi$. The lines are linear regressions used to extrapolate $\langle \chi \rangle$. The dependence on the substrate dimension of the universal (c) cumulants and (d) exponents. The lines are power-law regressions. The roughness exponents were taken from Ref. [26].

Figure 5

Density probability distribution for the RSOS model at substrates with dimensions $d=1$–4. The case $d=1$ was not rescaled by the factor $2^{1/3}$ to render the Gaussian orthogonal ensemble (GOE) distribution [6].