zbMATH — the first resource for mathematics

Gaussian limits for multidimensional random sequential packing at saturation. (English) Zbl 1145.60017
Summary: 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 \(\lambda\) is asymptotically normal as \(\lambda \rightarrow \infty\). 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.

60F05 Central limit and other weak theorems
60D05 Geometric probability and stochastic geometry
Full Text: DOI arXiv
[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 · doi:10.1016/0001-8686(94)80008-1
[2] Baryshnikov Yu. and Yukich J.E. (2003). Gaussian fields and random packing. J. Stat. Phys. 111: 443–463 · Zbl 1033.60060 · doi:10.1023/A:1022229713275
[3] Baryshnikov Yu. and Yukich J.E. (2005). Gaussian limits for random measures in geometric probability. Annals Appl. Prob. 15: 213–253 · Zbl 1068.60028 · doi:10.1214/105051604000000594
[4] Bartelt M.C. and Privman V. (1991). Kinetics of irreversible monolayer and multilayer sequential adsorption. Internat. J. Mod. Phys. B 5: 2883–2907 · doi:10.1142/S0217979291001127
[5] Coffman E.G., Flatto L. and Jelenković P. (2000). Interval packing: the vacant interval distribution. Annals of Appl. Prob. 10: 240–257 · Zbl 1161.60338 · doi:10.1214/aoap/1019737671
[6] Coffman E.G., Flatto L., Jelenković P. and Poonen B. (1998). Packing random intervals on-line. Algorithmica 22: 448–476 · Zbl 0914.68082 · doi:10.1007/PL00009233
[7] Diggle, P.J.: Statistical Analysis of Spatial Point Patterns. London: Academic Press 1983 · Zbl 0559.62088
[8] Dvoretzky, A., Robbins, H.: On the ”parking” problem. MTA Mat Kut. Int. Kzl., (Publications of the Math. Res. Inst. of the Hungarian Academy of Sciences), 9, 209–225 (1964) · Zbl 0251.60023
[9] Evans J.W. (1993). Random and cooperative adsorption. Rev. Mod. Phys. 65: 1281–1329 · doi:10.1103/RevModPhys.65.1281
[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 · doi:10.1063/1.1733154
[12] Penrose M.D. (2001). Random parking, sequential adsorption and the jamming limit. Commun. Math. Phys. 218: 153–176 · Zbl 0980.60020 · doi:10.1007/s002200100387
[13] Penrose M.D. (2001). Limit theorems for monolayer ballistic deposition in the continuum. J. Stat. Phys. 105: 561–583 · Zbl 0990.60033 · doi:10.1023/A:1012275725505
[14] Penrose M.D. (2005). Multivariate spatial central limit theorems with applications to percolation and spatial graphs. Ann. Prob. 33: 1945–1991 · Zbl 1087.60022 · doi:10.1214/009117905000000206
[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 · Zbl 1018.60023 · doi:10.1214/aoap/1015961164
[18] Penrose M.D. and Yukich J.E. (2003). Weak laws of large numbers in geometric probability. Ann. Appl. Probab. 13: 277–303 · Zbl 1029.60008 · doi:10.1214/aoap/1042765669
[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. Kzl., (Publications of the Math. Res. Inst. of the Hungarian Academy of Sciences) 3, 109–127 (1958) · Zbl 0105.11903
[22] Quintanilla J. and Torquato S. (1997). Local volume fluctuations in random media. J. Chem. Phys. 106: 2741–2751 · doi:10.1063/1.473414
[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 · Zbl 1145.60017
[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 · Zbl 1073.60022 · doi:10.1016/j.spa.2005.03.007
[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 · doi:10.1016/S0927-7757(99)00409-4
[26] Torquato, S.: Random Heterogeneous Materials, Springer Interdisciplinary Applied Mathematics, New York: Springer-Verlag 2002 · Zbl 0988.74001
[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 · doi:10.1103/PhysRevE.74.061308
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.