# Growth Rate and Ergodicity Conditions for a Class of Random Trees

Abstract : This paper gives the growth rate and the ergodicity conditions for a simple class of random trees. New edges appear according to a Poisson process, and leaves can be deleted at a rate $\mu$. The main results lay the stress on the famous number $e$. In the case of a pure birth process, i.e. $\mu=0$, the height of the tree at time $t$ grows linearly at the rate $e$, in mean and almost surely as $t\to\infty$. When deletions of leaves are permitted, a complete classification of the process is given in terms of the intensity factor $\rho=\lambda/\mu\,$: it is ergodic if $\rho\leq e^{-1}$, and transient if $\rho>e^{-1}$. 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. Bounds are obtained for the transient regime.
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https://hal.inria.fr/inria-00072256
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Submitted on : Tuesday, May 23, 2006 - 8:14:20 PM
Last modification on : Friday, May 25, 2018 - 12:02:05 PM
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### Identifiers

• HAL Id : inria-00072256, version 1

### Citation

Guy Fayolle, Maxim Krikun. Growth Rate and Ergodicity Conditions for a Class of Random Trees. [Research Report] RR-4331, INRIA. 2001. ⟨inria-00072256⟩

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