On the distribution of surface extrema in several one- and two-dimensional random landscapes

TL;DR

This paper analyzes the distribution of local maxima in one- and two-dimensional random surface growth models, providing exact results in 1D and Gaussian approximations in 2D, linking growth processes to permutation patterns.

Contribution

It establishes a novel mapping between ballistic growth surfaces and permutation sequences, enabling exact and approximate analysis of surface extrema distributions.

Findings

Exact distribution of maxima in 1D surfaces

Gaussian limit for 2D surface maxima distribution

Connection between growth models and permutation patterns

Abstract

We study here a standard next-nearest-neighbor (NNN) model of ballistic growth on one- and two-dimensional substrates focusing our analysis on the probability distribution function $P(M,L)$ of the number $M$ of maximal points (i.e., local ``peaks'') of growing surfaces. Our analysis is based on two central results: (i) the proof (presented here) of the fact that uniform one--dimensional ballistic growth process in the steady state can be mapped onto ''rise-and-descent'' sequences in the ensemble of random permutation matrices; and (ii) the fact, established in Ref. \cite{ov}, that different characteristics of ``rise-and-descent'' patterns in random permutations can be interpreted in terms of a certain continuous--space Hammersley--type process. For one--dimensional system we compute $P(M,L)$ exactly and also present explicit results for the correlation function characterizing the enveloping surface. For surfaces grown on 2d substrates, we pursue similar approach considering the ensemble of permutation matrices with long--ranged correlations. Determining exactly the first three cumulants of the corresponding distribution function, we define it in the scaling limit using an expansion in the Edgeworth series, and show that it converges to a Gaussian function as $L \to \infty$.