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


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 → ∞.


ballistic growth distribution of extremal points random permutation Eulerian random walk 


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  1. M. Kardar, G. Parisi and Y.-C. Zhang, Phys. Rev. Lett. 56:889 (1986).CrossRefADSzbMATHGoogle Scholar
  2. S. F. Edwards and D. R. Wilkinson, Proc. Roy. Soc. London A 381:17 (1982).ADSMathSciNetCrossRefGoogle Scholar
  3. S. N. Majumdar and A. Comtet, Phys. Rev. Lett. 92:225501 (2004).CrossRefADSGoogle Scholar
  4. M. A. Herman and H. Sitter, Molecular beam epitaxy: Fundamentals and current, (Springer: Berlin, 1996).Google Scholar
  5. P. Meakin, Fractals, Scaling, and Growth Far From Equilibrium, (Cambridge University Press: Cambridge, 1998).zbMATHGoogle Scholar
  6. M. Prähofer and H. Spohn, Phys. Rev. Lett. 84:4882 (2000).CrossRefADSGoogle Scholar
  7. J. Baik and E. M. Rains, J. Stat. Phys. 100:523 (2000).CrossRefMathSciNetzbMATHGoogle Scholar
  8. K. Johansson, Comm. Math. Phys. 242:277 (2003).ADSMathSciNetzbMATHGoogle Scholar
  9. C. A. Tracy and H. Widom, Commun. Math. Phys. 159:151 (1994).CrossRefADSMathSciNetzbMATHGoogle Scholar
  10. B. B. Mandelbrot, The fractal Geometry of nature. Freeman, New York, 1982.zbMATHGoogle Scholar
  11. P. Meakin, P. Ramanlal, L. M. Sander and R. C. Ball, Phys. Rev. A 34:5091 (1986).CrossRefADSGoogle Scholar
  12. J. Krug and P. Meakin, Phys. Rev. A 40:2064 (1989).CrossRefADSGoogle Scholar
  13. D. Blomker, S. Maier-Paape and T. Wanner, Interfaces and Free Boundaries 3:465 (2001).MathSciNetGoogle Scholar
  14. G. Costanza, Phys. Rev. E 55:6501 (1997).CrossRefADSGoogle Scholar
  15. F. D. A. Aarao Reis, Phys. Rev. E 63:056116 (2001).CrossRefADSGoogle Scholar
  16. E. Katzav and M. Schwartz, Phys. Rev. E 70:061608 (2004).CrossRefADSMathSciNetGoogle Scholar
  17. B. Derrida, private communication.Google Scholar
  18. S. Nechaev and R. Bikbov, Phys. Rev. Lett. 87:150602 (2001).CrossRefGoogle Scholar
  19. Z. Toroczkai, G. Korniss, S. Das Sarma and R. K. P. Zia, Phys. Rev. E 62:276 (2000).CrossRefADSGoogle Scholar
  20. J. Desbois, J. Phys. A: Math. Gen. 34:1953 (2001).CrossRefMathSciNetADSGoogle Scholar
  21. J. R. Stembrige, Trans. Am. Math. Soc. 349:763 (1997).CrossRefGoogle Scholar
  22. N. Bergeron, F. Hivert and J. -Y. Thibon, J. Combinatorial Theory A 117(1): (2004).Google Scholar
  23. L. J. Billera, S. K.Hsiao and S. Van Willigenburg, Adv. Math. 176:248 (2003).CrossRefMathSciNetzbMATHGoogle Scholar
  24. G. Oshanin and R. Voituriez, J. Phys. A: Math. Gen. 37:6221 (2004).CrossRefADSMathSciNetzbMATHGoogle Scholar
  25. H. Cramér, Mathematical methods of Statistics, (Princeton University Press: Princeton, 1957).Google Scholar
  26. Only the particle of the roof T can be removed from the pile without disturbing the rest of the heap—as in the micado game.Google Scholar
  27. B. Derrida and E. Gardner, J. Phys. (Paris) 47:959 (1986).Google Scholar
  28. J. M. Hammersley, Proc. 6th Berkeley Symp. Math. Statist. and Probability, Vol. 1. (Berkeley, CA: University of California Press, 1972), p. 345.Google Scholar
  29. D. Aldous and P. Diaconis, Probab. Theory Relat. Fields 103:199 (1995).CrossRefMathSciNetzbMATHGoogle Scholar

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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