Skip to main content
SpringerLink
Log in

Gaussian Limits for Multidimensional Random Sequential Packing at Saturation

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Consider the random sequential packing model with infinite input and in any dimension. When the input consists of non-zero volume convex solids we show that the total number of solids accepted over cubes of volume λ is asymptotically normal as λ → ∞. We provide a rate of approximation to the normal and show that the finite dimensional distributions of the packing measures converge to those of a mean zero generalized Gaussian field. The method of proof involves showing that the collection of accepted solids satisfies the weak spatial dependence condition known as stabilization.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adamczyk Z., Siwek B., Zembala M. and Belouschek P. (1994). Kinetics of localized adsorption of colloid particles. Adv. in Colloid and Interface Sci. 48: 151–280

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  3. Baryshnikov Yu. and Yukich J.E. (2005). Gaussian limits for random measures in geometric probability. Annals Appl. Prob. 15: 213–253

    Article  MATH  MathSciNet  Google Scholar 

  4. Bartelt M.C. and Privman V. (1991). Kinetics of irreversible monolayer and multilayer sequential adsorption. Internat. J. Mod. Phys. B 5: 2883–2907

    Article  ADS  MathSciNet  Google Scholar 

  5. Coffman E.G., Flatto L. and Jelenković P. (2000). Interval packing: the vacant interval distribution. Annals of Appl. Prob. 10: 240–257

    Article  MATH  Google Scholar 

  6. Coffman E.G., Flatto L., Jelenković P. and Poonen B. (1998). Packing random intervals on-line. Algorithmica 22: 448–476

    Article  MATH  MathSciNet  Google Scholar 

  7. Diggle, P.J.: Statistical Analysis of Spatial Point Patterns. London: Academic Press 1983

  8. Dvoretzky, A., Robbins, H.: On the “parking” problem. MTA Mat Kut. Int. Kz̈l., (Publications of the Math. Res. Inst. of the Hungarian Academy of Sciences), 9, 209–225 (1964)

  9. Evans J.W. (1993). Random and cooperative adsorption. Rev. Mod. Phys. 65: 1281–1329

    Article  ADS  Google Scholar 

  10. Grimmett, G.: Percolation, Second Edition, Berlin: Springer 1999

  11. Mackenzie J.K. (1962). Sequential filling of a line by intervals placed at random and its application to linear adsorption. J. Chem. Phys. 37(4): 723–728

    Article  Google Scholar 

  12. Penrose M.D. (2001). Random parking, sequential adsorption and the jamming limit. Commun. Math. Phys. 218: 153–176

    Article  MATH  ADS  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  14. Penrose M.D. (2005). Multivariate spatial central limit theorems with applications to percolation and spatial graphs. Ann. Prob. 33: 1945–1991

    Article  MATH  MathSciNet  Google Scholar 

  15. Penrose, M.D.: Laws of large numbers for random measures in geometric probability Preprint, 2005

  16. Penrose, M.D.: Gaussian limits for random geometric measures Preprint, 2005

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

    Article  MATH  MathSciNet  Google Scholar 

  18. Penrose M.D. and Yukich J.E. (2003). Weak laws of large numbers in geometric probability. Ann. Appl. Probab. 13: 277–303

    Article  MATH  MathSciNet  Google Scholar 

  19. Penrose, M.D., Yukich, J.E.: Normal approximation in geometric probability. In: Stein’s Method and Applications Lecture Note Series, Institute for Mathematical Sciences, National University of Singapore, 5, A.D. Barbour, Louis H.Y. Chen, eds., 2005, pp. 37–58. Also available electronically from http://arxiv.org/list/math.PR/0409088, 2004

  20. Privman V.: Adhesion of Submicron Particles on Solid Surfaces. In: A Special Issue of Colloids and Surfaces A 165, edited by V. Privman, 2000

  21. Rényi, A.: On a one-dimensional random space-filling problem, MTA Mat Kut. Int. Kz̈l., (Publications of the Math. Res. Inst. of the Hungarian Academy of Sciences) 3, 109–127 (1958)

  22. Quintanilla J. and Torquato S. (1997). Local volume fluctuations in random media. J. Chem. Phys. 106: 2741–2751

    Article  ADS  Google Scholar 

  23. Schreiber, T., Penrose, M.D., Yukich J.E.: Gaussian limits for multidimensional random sequential packing at saturation (extended version) http://arxiv.org/list/math.PR/0610680, 2006

  24. Schreiber T. and Yukich J.E. (2005). Large deviations for functionals of spatial point processes with applications to random packing and spatial graphs. Stochastic Processes and Their Applications 115: 1332–1356

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  26. Torquato, S.: Random Heterogeneous Materials, Springer Interdisciplinary Applied Mathematics, New York: Springer-Verlag 2002

  27. Torquato S., Uche O.U. and Stillinger F.H. (2006). Random sequential addition of hard spheres in high Euclidean dimensions. Phys. Rev. E 74: 061308

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Nicholas Copernicus University, Toruń, Poland

    T. Schreiber

  2. Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, United Kingdom

    Mathew D. Penrose

  3. Department of Mathematics, Lehigh University, Bethlehem, PA, 18015, USA

    J. E. Yukich

Authors
  1. T. Schreiber
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mathew D. Penrose
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. E. Yukich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mathew D. Penrose.

Additional information

Communicated by J.L. Lebowitz

Research partially supported by the Polish Minister of Scientific Research and Information Technology grant 1 P03A 018 28 (2005-2007)

Research supported in part by NSF grant DMS-0203720

Rights and permissions

Reprints and permissions

About this article

Cite this article

Schreiber, T., Penrose, M.D. & Yukich, J.E. Gaussian Limits for Multidimensional Random Sequential Packing at Saturation. Commun. Math. Phys. 272, 167–183 (2007). https://doi.org/10.1007/s00220-007-0218-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-007-0218-2

Keywords

Search

Navigation