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

Florent Hivert 1, S. Nechaev 2, 3, G. Oshanin 4, 5, 6, O. Vasilyev 4, 7

Journal of Statistical Physics 126 (2007) 243-279

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 $L \to \infty$.

  • 1. Laboratoire d'Informatique, de Traitement de l'Information et des Systèmes (LITIS),
    Institut National des Sciences Appliquées (INSA) - Rouen – Université du Havre – Université de Rouen : EA4108
  • 2. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
    CNRS : UMR8626 – Université Paris XI - Paris Sud
  • 3. P. N. Lebedev Physical Institute,
    Russian Academy of Science
  • 4. Laboratoire de Physique Théorique de la Matière Condensée (LPTMC),
    CNRS : UMR7600 – Université Paris VI - Pierre et Marie Curie
  • 5. Max-Planck-Institut fur Metallforschung,
  • 6. Institut fur Theoretische und Angewandte Physik,
    Universität Stuttgart
  • 7. Center for Molecular Modelling, Materia Nova,
    Université de Mons-Hainaut