We study here a standard next-nearest-neighbor (NNN) model of ballisticgrowth on one- and two-dimensional substrates focusing our analysis on theprobability distribution function $P(M,L)$ of the number $M$ of maximal points(i.e., local ``peaks'') of growing surfaces. Our analysis is based on twocentral results: (i) the proof (presented here) of the fact that uniformone--dimensional ballistic growth process in the steady state can be mappedonto ''rise-and-descent'' sequences in the ensemble of random permutationmatrices; and (ii) the fact, established in Ref. \cite{ov}, that differentcharacteristics of ``rise-and-descent'' patterns in random permutations can beinterpreted in terms of a certain continuous--space Hammersley--type process.For one--dimensional system we compute $P(M,L)$ exactly and also presentexplicit results for the correlation function characterizing the envelopingsurface. For surfaces grown on 2d substrates, we pursue similar approachconsidering the ensemble of permutation matrices with long--rangedcorrelations. Determining exactly the first three cumulants of thecorresponding distribution function, we define it in the scaling limit using anexpansion in the Edgeworth series, and show that it converges to a Gaussianfunction as $L \to \infty$.
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机译：我们在这里研究一维和二维衬底上弹道增长的标准近邻（NNN）模型，重点分析最大点数（M）的概率分布函数$ P（M，L）$ ，生长表面的局部``峰值''）。我们的分析基于两个中心结果：（i）证明以下事实的事实：稳态下的均匀一维弹道生长过程可以映射到随机集合中的“起落”序列排列矩阵（ii）参考文件中确立的事实。 \ cite {ov}，可以用某种连续的空间Hammersley型过程来解释随机排列中``上升和下降''模式的不同特征。对于一维系统，我们计算$ P（ M，L）$准确地给出了表征包络面的相关函数的明确结果。对于在2d基板上生长的表面，我们采用类似的方法，即考虑具有长程相关性的置换矩阵的集合。确切地确定相应分布函数的前三个累积量，我们使用Edgeworth系列的展开式在缩放极限中定义它，并证明它收敛为高斯函数，即$ L \ to \ infty $。
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