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

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 $β>0$. 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 $β(n-k)/n$, 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 $T_n = β^{-1} n \ln(n /\ln 4) + (\ln 2)/(2 β)$. We prove that $2^{-n} Z_n(T_n + n τ)$, $τ\in\bb R$, converges to the Gompertz curve $\exp (- (\ln 2) e^{-βτ})$. We also prove a central limit theorem for the martingale associated to $Z_n(t)$.

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