Abstract.
The main substance of the paper concerns the growth rate and the classification (ergodicity, transience) of a family of random trees. In the basic model, new edges appear according to a Poisson process of parameter λ and leaves can be deleted at a rate μ. The main results lay the stress on the famous number e. A complete classification of the process is given in terms of the intensity factor ρ=λ/μ: it is ergodic if ρ≤e −1, and transient if ρ>e −1. There is a phase transition phenomenon: the usual region of null recurrence (in the parameter space) here does not exist. This fact is rare for countable Markov chains with exponentially distributed jumps. Some basic stationary laws are computed, e.g. the number of vertices and the height. Various bounds, limit laws and ergodic-like theorems are obtained, both for the transient and ergodic regimes. In particular, when the system is transient, the height of the tree grows linearly as the time t→∞, at a rate which is explicitly computed. Some of the results are extended to the so-called multiclass model.
References
De Bruijn, N.G.: Asymptotic Methods in Analysis. North-Holland, second edition, 1961
Cartan, H.: Cours de calcul différentiel. Hermann, Collection Méthodes, 1977
Delcoigne, F., Fayolle, G.: Thermodynamical limit and propagation of chaos in polling systems. Markov Processes and Related Fields 5 (1), 89–124 (1999)
Devroye, L.: Branching processes in the analysis of the height of trees. Acta Informatica 24, 277–298 (1987)
Fayolle, G., Krikun, M.: Growth rate and ergodicity conditions for a class of random trees. Mathematics and Computer Science II, Birkhaüser Verlag Basel/Switzerland, 2002
Feller, W.: An Introduction to Probability Theory and its Applications. Vol. I and II, Wiley, 1971
Fuchs, B.A., Levin, V.I.: Functions of a Complex Variable. Vol. II, Pergamon Press, 1961
Gantmacher, F.R.: The Theory of Matrices. Vol. II, Chelsea Publishing Company, 1960
Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Academic Press, corrected and enlarged edition, 1980
Kallenberg, O.: Foundations of Modern Probability. Springer, Probability and its Applications, 2001
Krikun, M.: Height of a random tree. Markov Processes and Related Fields 6 (2), 135–146 (2000)
Kuczma, M.: Functional Equations in a Single Variable. Polska Akademia Nauk, 46, Warszawa, 1968
Liggett, T.M.: Monotonicity of conditional distributions and growth models on trees. Ann. Prob. 28 (4), 1645–1665 (2000)
Mahmoud, H.M.: Evolution of Random Search Trees. Wiley-Intersciences Series, 1992
Pittel, B.: Note on the heights of random recursive trees and random m-ary search trees Random Structures and Algorithms. 5, 337–347 (1994)
Puha, A.L.: A reversible nearest particle system on the homogeneous tree. J. Theor. Prob. 12 (1), 217–253 (1999)
Sedgewick, R., Flajolet, Ph.: An Introduction to the Analysis of Algorithms. Addison-Wesley, 1996
Author information
Authors and Affiliations
Corresponding author
Additional information
J.-M. Lasgouttes worked partly on the present study while spending a sabbatical at EURANDOM in Eindhoven.
Mathematics Subject Classification (2000): 60B05, 60J80, 34L30
Revised: 2003
Rights and permissions
About this article
Cite this article
Fayolle, G., Krikun, M. & Lasgouttes, JM. Birth and death processes on certain random trees: classification and stationary laws. Probab. Theory Relat. Fields 128, 386–418 (2004). https://doi.org/10.1007/s00440-003-0311-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-003-0311-1
Keywords
- Random trees
- Ergodicity
- Transience
- Nonlinear differential equations
- Phase transition