Birth and death processes on certain random trees: Classification and stationary laws

TL;DR

This paper classifies the growth and long-term behavior of a family of random trees based on edge addition and leaf deletion rates, revealing a phase transition at a critical ratio related to the mathematical constant e.

Contribution

It provides a complete classification of the process's ergodic and transient regimes and computes key stationary laws, highlighting a rare phase transition phenomenon.

Findings

Process is ergodic if ; transient if ; no null recurrence region.

Height grows linearly in the transient regime, with an explicitly computed rate.

Stationary laws for vertex count and height are derived.

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 $\lambda$ and leaves can be deleted at a rate $\mu$. 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 $\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. 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 $t\to\infty$, at a rate which is explicitly computed. Some of the results are extended to the so-called multiclass model.