Growth Rate and Ergodicity Conditions for a Class of Random Trees

Abstract

The main substance of the paper concerns growth rate and classification (ergodicity, transience) of a family of random trees. In the basic model new edges appear according to a Poisson process of parameter A 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⁻¹and 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 A theorem, much of ergodic type is derived for the height of the tree at time t, which in the transient case is shown to grow linearly as t → ∞, at a rate explicitly computed.