Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T11:37:18.307Z Has data issue: false hasContentIssue false
  • English
  • Français

On a deposition process on the circle with disorder

Part of: Stochastic processes Distribution theory Combinatorial probability Special processes

Published online by Cambridge University Press:  01 July 2016

Thierry Huillet
Thierry Huillet*
Affiliation:
CNRS and Université de Cergy-Pontoise
*
∗ Postal address: Laboratoire de Physique Théorique et Modélisation, CNRS-UMR 8089 et Université de Cergy-Pontoise, 5 mail Gay-Lussac, 95031 Neuville sur Oise, France. Email address: thierry.huillet@ptm.u-cergy.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

Throw n points sequentially and at random onto a unit circle and append a clockwise arc (or rod) of length s to each such point. The resulting random set (the free gas of rods) is a union of a random number of clusters with random sizes modelling a free deposition process on a one-dimensional substrate. A variant of this model is investigated in order to take into account the role of the disorder, θ > 0; this involves Dirichlet(θ) distributions. For such free deposition processes with disorder θ, we shall be interested in the occurrence times and probabilities, as n grows, of two specific types of configurations: those avoiding overlapping rods (the hard-rod gas) and those for which the largest gap is smaller than the rod length s (the packing gas). Special attention is paid to the thermodynamic limit when ns = ρ for some finite density ρ of points. The occurrence of parking configurations, those for which hard-rod and packing constraints are both fulfilled, is then studied. Finally, some aspects of these problems are investigated in the low-disorder limit θ ↓ 0 as n ↑ ∞ while nθ = γ > 0. Here, Poisson-Dirichlet(γ) partitions play some role.

Keywords

Random covering of the circlerandom growthhard-rod configurationpacking configurationparking configurationfree deposition process with disorder θDirichlet and Poisson-Dirichlet distributions with low disorder

MSC classification

Primary: 60C05: Combinatorial probability 60G57: Random measures 62E17: Approximations to distributions (nonasymptotic)
Secondary: 60K99: None of the above, but in this section 62E15: Exact distribution theory 62E20: Asymptotic distribution theory
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 36 , Issue 4 , December 2004 , pp. 996 - 1020
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Carlton, M. (1999). Applications of the two-parameter Poisson–Dirichlet distribution. , University of California, Los Angeles.Google Scholar
[2] Darling, D. A. (1953). On a class of problems related to the random division of an interval. Ann. Math. Statist. 24, 239–253.Google Scholar
[3] De Coninck, J., Dunlop, F. and Rivasseau, V. (1989). On the microscopic validity of the Wulff construction and of the generalized Young equation. Commun. Math. Phys. 121, 401–419.Google Scholar
[4] Domb, C. (1989). Covering by random intervals and one-dimensional continuum percolation. J. Statist. Phys. 55, 441–460.Google Scholar
[5] Dunlop, F. and Huillet, T. (2003). Hard rods: statistics of parking configurations. Physica A 324, 698–706.CrossRefGoogle Scholar
[6] Evans, J. W. (1993). Random and cooperative sequential adsorption. Rev. Modern Phys. 65, 1281–1329.Google Scholar
[7] Fisher, R. A. (1929). Tests of significance in harmonic analysis. Proc. R. Soc. London A 125, 54–59.Google Scholar
[8] Flatto, L. (1973). A limit theorem for random coverings of a circle. Israel J. Math. 15, 167–184.Google Scholar
[9] Flatto, L. and Konheim, A. G. (1962). The random division of an interval and the random covering of a circle. SIAM Rev. 4, 211–222.Google Scholar
[10] Hemmer, P. C. (1989). The random parking problem. J. Statist. Phys. 57, 865–869.Google Scholar
[11] Holst, L. (1983). A note on random arcs on the circle. In Probability and Mathematical Statistics, eds Gut, A. and Holst, L., Department of Mathematics, Uppsala University, pp. 40–46.Google Scholar
[12] Holst, L. (2001). The Poisson–Dirichlet distribution and its relatives revisited. Preprint, Royal Institute of Technology, Stockholm.Google Scholar
[13] Holst, L. and Hüsler, J. (1984). On the random coverage of the circle. J. Appl. Prob. 21, 558–566.Google Scholar
[14] Huillet, T. (2003). Random covering of the circle: the size of the connected components. Adv. Appl. Prob. 35, 563–582.Google Scholar
[15] Huillet, T. (2003). Random covering of the circle: the configuration space of the free deposition process. J. Phys. A. 36, 12143–12155.Google Scholar
[16] Huillet, T. and Martínez, S. (2003). Sampling from finite random partitions. Methodol. Comput. Appl. Prob. 5, 467–492.Google Scholar
[17] Jeulin, D. (1997). Dead leaves models: from space tessellation to random functions. In Proceedings of the International Symposium on Advances in Theory and Applications of Random Sets (Fontainebleau, 1996), World Scientific, River Edge, NJ, pp. 137–156.Google Scholar
[18] Kingman, J. F. C. (1993). Poisson Processes. Oxford University Press.Google Scholar
[19] Lévy, P., (1939). Sur la division d'un segment par des points choisis au hasard. C. R. Acad. Sci. Paris 208, 147–149.Google Scholar
[20] Meeron, E. and Siegert, A. J. F. (1968). Statistical mechanics of hard particle systems. J. Chem. Phys. 48, 3139–3155.Google Scholar
[21] Pitman, J. (1996). Random discrete distributions invariant under size-biased permutation. Adv. Appl. Prob. 28, 525–539.CrossRefGoogle Scholar
[22] Pitman, J. and Yor, M. (1997). The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator. Ann. Prob. 25, 855–900.Google Scholar
[23] Pyke, R. (1965). Spacings (with discussion). J. R. Statist. Soc. B 27, 395–449.Google Scholar
[24] Rényi, A., (1958). On a one-dimensional problem concerning random space filling. Magyar Tud. Akad. Mat. Kutató Int. Közl. 3, 109–127.Google Scholar
[25] Siegel, A. F. (1978). Random arcs on the circle. J. Appl. Prob. 15, 774–789.Google Scholar
[26] Steutel, F. W. (1967). Random division of an interval. Statist. Neerlandica 21, 231–244.Google Scholar
[27] Stevens, W. L. (1939). Solution to a geometrical problem in probability. Ann. Eugenics 9, 315–320.Google Scholar
[28] Weiss, L. (1959). The limiting joint distribution of the largest and smallest sample spacings. Ann. Math. Statist. 30, 590–593.Google Scholar
[29] Whitworth, W. A. (1897). DCC Exercises on Choice and Chance. Deighton Bell and Co., Cambridge. Republished: Hafner, New York, 1959.Google Scholar
[30] Widom, B. (1973). Random sequential filling of intervals on a line. J. Chem. Phys. 58, 4043–4044.Google Scholar