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On Ballistic Deposition Process on a Strip

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Abstract

We revisit the model of the ballistic deposition studied in Atar et al. (Electron Commun Probab 6:31–38, 2001) and prove several combinatorial properties of the random tree structure formed by the underlying stochastic process. Our results include limit theorems for the number of roots and the empirical average of the distance between two successive roots of the underlying tree-like structure as well as certain intricate moments calculations.

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References

  1. Page, E.S.: The distribution of vacancies on a line. J. R. Stat. Soc. Ser. B 21, 364–374 (1959)

    ADS  MathSciNet  MATH  Google Scholar 

  2. Rényi, A.: A one-dimensional problem concerning random space-filling. Magyar Tud. Akad. Mat. Kutató Int. Közl. 3, 109–127 (1958)

    MathSciNet  MATH  Google Scholar 

  3. Cadilhe, A., Araújo, N.A.M., Privman, V.: Random sequential adsorption: from continuum to lattice and pre-patterned substrates. J. Phys. Condens. Matter 19, 065124 (2007)

    Article  ADS  Google Scholar 

  4. Evans, J.W.: Random and cooperative adsorption. Rev. Modern Phys. 65, 1281–1329 (1993)

    Article  ADS  Google Scholar 

  5. Penrose, M.D., Yukich, J.E.: Limit theory for random sequential packing and deposition. Ann. Appl. Probab. 12, 272–301 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Sikiri, M.D., Itoh, Y.: Random Sequential Packing of Cubes. World Scientific, Singapore (2011)

    Book  Google Scholar 

  7. Talbot, J., Tarjus, G., Van Tassel, P.R., Viot, P.: From car parking to protein adsorption: an overview of sequential adsorption processes. Colloid Surf. A 165, 287–324 (2000)

    Article  Google Scholar 

  8. Baule, A.: Shape universality classes in the random sequential adsorption of nonspherical particles. Phys. Rev. Lett. 119, 028003 (2017)

    Article  ADS  Google Scholar 

  9. Baule, A.: Optimal random deposition of interacting particles, Phys. Rev. Lett., to appear. arXiv:1903.02101

  10. Cieśla, M., Paja̧k, G., Ziff, R.M.: In a search for a shape maximizing packing fraction for two-dimensional random sequential adsorption. J. Chem. Phys. 145, 044708 (2016)

    Article  ADS  Google Scholar 

  11. Cieśla, M., Kubala, P.: Random sequential adsorption of cubes. J. Chem. Phys. 148, 024501 (2018)

    Article  ADS  Google Scholar 

  12. Clay, M., Simányi, N.: Rényi’s parking problem revisited. Stoch. Dyn. 16, 1660006 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Krapivsky, P.L., Luck, J.M.: Coverage fluctuations in theater models. arXiv:1902.04365

  14. Barabasi, A., Stanley, H.E.: Fractal Concepts in Surface Growth. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  15. Family, F., Vicsek, T. (eds.): Dynamics of Fractal Surfaces. World Scientific, Singapore (1991)

    MATH  Google Scholar 

  16. Robledo, A., Grabill, C.N., Kuebler, S.M., Dutta, A., Heinrich, H., Bhattacharya, A.: Morphologies from slippery ballistic deposition model: a bottom-up approach for nanofabrication. Phys. Rev. E 83, 051604 (2011)

    Article  ADS  Google Scholar 

  17. Schaaf, P., Voegel, J.-C., Senger, B.: From random sequential adsorption to ballistic deposition: a general view of irreversible deposition processes. J. Phys. Chem. B 104, 2204–2214 (2000)

    Article  Google Scholar 

  18. Vold, M.J.: A numerical approach to the problem of sediment volume. J. Colloid Sci. 14, 168–174 (1959)

    Article  Google Scholar 

  19. Sutherland, D.N.: Comments on Vold’s simulation of floc formation. J. Colloid Sci. 22, 300–302 (1966)

    Article  ADS  Google Scholar 

  20. Giri, A., Tarafdar, S., Gouze, P., Dutta, T.: Fractal pore structure of sedimentary rocks: simulation in \(2\)-\(D\) using a relaxed disperse ballistic deposition model. J. Appl. Geophys. 87, 40–45 (2012)

    Article  ADS  Google Scholar 

  21. Forgerini, F.L., Marchiori, R.: A brief review of mathematical models of thin film growth and surfaces: a possible route to avoid defects in stents. Biomatter 4, e28871 (2014)

    Article  Google Scholar 

  22. Meakin, P., Jullien, R.: Invited paper. Simple ballistic deposition models for the formation of thin films. In: Jacobson, M.R. (ed.) Modeling of Optical Thin Films, vol. 821, pp. 45–56. International Society for Optics and Photonics, Bellingham (1988)

    Chapter  Google Scholar 

  23. Privman, V. (Ed.), Collection of review articles: Adhesion of submicron particles on solid surfaces, Colloids Surf. A 165, special issue (2000)

  24. Costa, M., Menshikov, M., Shcherbakov, V., Vachkovskaia, M.: Localisation in a growth model with interaction. J. Stat. Phys. 171, 1150–1175 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Menshikov, M., Shcherbakov, V.: Localisation in a growth model with interaction. Arbitrary graphs. arXiv:1903.04418

  26. Shcherbakov, V., Volkov, S.: Stability of a growth process generated by monomer filling with nearest-neighbour cooperative effects. Stoch. Process. Appl. 120, 926–948 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Amar, J.G., Family, F.: Phase transition in a restricted solid-on-solid surface-growth model in \(2+ 1\) dimensions. Phys. Rev. E 64, 543 (1990)

    ADS  Google Scholar 

  28. Aarão Reis, F.D.A.: Universality and corrections to scaling in the ballistic deposition model. Phys. Rev. E 63, 056116 (2001)

    Article  ADS  Google Scholar 

  29. Haselwandter, C.A., Vvedensky, D.D.: Scaling of ballistic deposition from a Langevin equation. Phys. Rev. E 73, 040101 (2006)

    Article  ADS  Google Scholar 

  30. Katzav, E., Schwartz, M.: What is the connection between ballistic deposition and the Kardar–Parisi–Zhang equation? Phys. Rev. E 70, 061608 (2004)

    Article  ADS  Google Scholar 

  31. Majumdar, S.N., Nechaev, S.: Anisotropic ballistic deposition model with links to the Ulam problem and the Tracy–Widom distribution. Phys. Rev. E 69, 011103 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  32. Nagatani, T.: From ballistic deposition to the Kardar–Parisi–Zhang equation through a limiting procedure. Phys. Rev. E 58, 700 (1998)

    Article  ADS  Google Scholar 

  33. D’Souza, R.M.: Anomalies in simulations of nearest neighbor ballistic deposition. Int. J. Mod. Phys. C 8, 941–951 (1997)

    Article  ADS  Google Scholar 

  34. Kartha, M.J.: Surface morphology of ballistic deposition with patchy particles and visibility graph. Phys. Lett. A 381, 556–560 (2016)

    Article  ADS  Google Scholar 

  35. Kwak, W., Kim, J.M.: Random deposition model with surface relaxation in higher dimensions. Physica A 520, 87–92 (2019)

    Article  ADS  Google Scholar 

  36. Mal, B., Ray, S., Shamanna, J.: Surface properties and scaling behavior of a generalized ballistic deposition model. Phys. Rev. E 93, 022121 (2016)

    Article  ADS  Google Scholar 

  37. Oliveira Filho, J.S., Oliveira, T.J., Redinz, J.A.: Surface and bulk properties of ballistic deposition models with bond breaking. Physica A 392, 2479–2486 (2013)

    Article  ADS  Google Scholar 

  38. Penrose, M.D.: Growth and roughness of the interface for ballistic deposition. J. Stat. Phys. 131, 247–268 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. Penrose, M.D.: Existence and spatial limit theorems for lattice and continuum particle systems. Probab. Surv. 5, 1–36 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  40. Seppäläinen, T.: Strong law of large numbers for the interface in ballistic deposition. Ann. Inst. H. Poincaré Probab. Statist. 36, 691–736 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. Corwin, I.: Kardar–Parisi–Zhang universality. Not. Am. Math. Soc. 63, 230–239 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  42. Atar, R., Athreya, S., Kang, M.: Ballistic deposition on a planar strip. Electron. Commun. Probab. 6, 31–38 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  43. Asselah, A., Cirillo, E.N.M., Scoppola, B., Scoppola, E.: On diffusion limited deposition. Electron. J. Probab. 21, 1–29 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  44. Talbot, J., Ricci, S.M.: Analytic model for a ballistic deposition process. Phys. Rev. Lett. 68, 958–962 (1992)

    Article  ADS  Google Scholar 

  45. Penrose, M.D.: Limit theorems for monolayer ballistic deposition in the continuum. J. Stat. Phys. 105, 561–583 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  46. Coulon-Prieur, C., Doukhan, P.: A triangular central limit theorem under a new weak dependence condition. Stat. Probab. Lett. 47, 61–68 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  47. Neumann, M.H.: A central limit theorem for triangular arrays of weakly dependent random variables, with applications in statistics. ESAIM Probab. Stat. 17, 120–134 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  48. Peligrad, M.: On the central limit theorem for triangular arrays of \(\phi \)-mixing sequences. In: Szyszkowicz, B. (ed.) Asymptotic Methods in Probability and Statistics. A Volume in Honour of Miklós Csörgő, pp. 49–55. North-Holland, Amsterdam (1998)

    Chapter  Google Scholar 

  49. Baryshnikov, Yu., Yukich, J.E.: Gaussian fields and random packing. J. Stat. Phys. 111, 443–463 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  50. Hwang, H.K.: Large deviations for combinatorial distributions. I. Central limit theorems. Ann. Appl. Probab. 6, 297–319 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  51. Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2008)

    MATH  Google Scholar 

  52. Hwang, H.K.: Large deviations of combinatorial distributions. II. Local limit theorems. Ann. Appl. Probab. 8, 163–181 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  53. Hille, E.: Ordinary Differential Equations in the Complex Domain. Wiley, New York (1976)

    MATH  Google Scholar 

  54. Janson, S.: Normal convergence by higher semi-invariants with applications to sums of dependent random variables and random graphs. Ann. Probab. 16, 305–312 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  55. Rinott, Y.: On normal approximation rates for certain sums of dependent random variables. J. Comput. Appl. Math. 55, 135–143 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  56. Als-Nielsen, J., Birgeneau, R.J.: Mean field theory, the Ginzburg criterion, and marginal dimensionality of phase transitions. Am. J. Phys. 45, 554–560 (1977)

    Article  ADS  Google Scholar 

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Acknowledgements

We are grateful to the referee for comments and feedback on the earlier version of the manuscript that resulted in a better presentation of results and proofs.

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Authors and Affiliations

  1. Department of Mathematics, University of Haifa, 199 Abba Khoushy Ave, 3498838, Haifa, Israel

    Toufik Mansour

  2. Occidental Petroleum Corporation, Houston, TX, 77046, USA

    Reza Rastegar

  3. Departments of Mathematics and Petroleum Engineering, University of Tulsa, Tulsa, OK, 74104, USA

    Reza Rastegar

  4. Department of Statistics, Texas A&M University, College Station, TX, 77843, USA

    Alexander Roitershtein

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  1. Toufik Mansour
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  2. Reza Rastegar
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  3. Alexander Roitershtein
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Correspondence to Reza Rastegar.

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Communicated by Eric A. Carlen.

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Mansour, T., Rastegar, R. & Roitershtein, A. On Ballistic Deposition Process on a Strip. J Stat Phys 177, 626–650 (2019). https://doi.org/10.1007/s10955-019-02383-4

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