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We investigate numerically the effects of long-range temporal and spatialcorrelations based on probability distribution of interface width $W(L,t)$within the early growth regimes in the (1+1)-dimensional Kardar-Parisi-Zhang(KPZ) growth system. Through extensive numerical simulations, we find thatlong-range temporally correlated noise could not significantly impact thedistribution form of interface width. Generally, $W(L,t)$ obeys Tracy-WidomGaussian symplectic ensembles(TW-GSE) when the temporal correlation exponent$\theta \ge 0$. On the other hand, the effects of long-range spatiallycorrelated noise are evidently different from the temporally correlated case.Our results show that, when the spatial correlation exponent $\rho \le 0.20$,the distribution form of $W(L,t)$ approaches to TW-GSE, and when $\rho > 0.20$,the distribution becomes more asymmetric, leptokurtic, and fat-tailed, andtends to an unknown form, which is similar to Gumbel distribution.

Probability distributions for kinetic roughening in the Kardar-Parisi-Zhang growth with long-range temporal and spatial correlations