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Journal of Statistical Physics

, Volume 126, Issue 2, pp 243–279 | Cite as

On the Distribution of Surface Extrema in Several One- and Two-dimensional Random Landscapes

  • F. Hivert
  • S. Nechaev
  • G. Oshanin
  • O. Vasilyev
Article

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. [G. Oshanin and R. Voituriez, J. Phys. A: Math. Gen. 37:6221 (2004)], 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 → ∞.

Keywords

ballistic growth distribution of extremal points random permutation Eulerian random walk 

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Copyright information

© Springer Science+Business Media, LLC 2006

Authors and Affiliations

  • F. Hivert
    • 1
  • S. Nechaev
    • 2
    • 4
  • G. Oshanin
    • 3
    • 5
    • 6
  • O. Vasilyev
    • 3
    • 7
  1. 1.LITIS/LIFARUniversité de RouenSaint Etienne du RouvrayFrance
  2. 2.LPTMSUniversité Paris SudOrsay CedexFrance
  3. 3.LPTMCUniversité Paris 6ParisFrance
  4. 4.P.N. Lebedev Physical Institute of the Russian Academy of SciencesMoscowRussia
  5. 5.Max-Planck-Institut für MetallforschungStuttgartGermany
  6. 6.Institut für Theoretische und Angewandte PhysikUniversität StuttgartStuttgartGermany
  7. 7.Center for Molecular Modelling, Materia NovaUniversity of Mons-HainautMonsBelgium

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