A Markovian growth dynamics on rooted binary trees evolving according to the Gompertz curve

Abstract

Inspired by biological dynamics, we consider a growth Markov process taking values on the space of rooted binary trees, similar to the Aldous-Shields model. Fix $n\ge 1$ and $\beta >0$. We start at time 0 with the tree composed of a root only. At any time, each node with no descendants, independently from the other nodes, produces two successors at rate $\beta (n-k)/n$, where $k$ is the distance from the node to the root. Denote by $Z_n(t)$ the number of nodes with no descendants at time $t$ and let $T_n={\beta }^{-1}n\ln (n/\ln 4)+(\ln 2)/(2\beta )$. We prove that $2^{-n}{Z}_n(T_n+n\tau )$, $τ\in\bb R$, converges to the Gompertz curve $\exp (-(\ln 2)e^{-\beta \tau })$. We also prove a central limit theorem for the martingale associated to $Z_n(t)$.

13 pages

References in corpus (2)

• Phase Transition in the Aldous-Shields Model of Growing Trees
• Gompertz mortality law and scaling behaviour of the Penna model