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On a deposition process on the circle with disorder

Part of: Stochastic processes Distribution theory Special processes Combinatorial probability

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
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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 

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