A Markovian Growth Dynamics on Rooted Binary Trees Evolving According to the Gompertz Curve
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Inspired by biological dynamics, we consider a growth Markov process taking values on the space of rooted binary trees, similar to the Aldous-Shields (Probab. Theory Relat. Fields 79(4):509–542, 1988) model. Fix n≥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 nln(n/ln4)+(ln2)/(2β). We prove that 2−n Z n (T n +nτ), τ∈ℝ, converges to the Gompertz curve exp(−(ln2) e −βτ ). We also prove a central limit theorem for the martingale associated to Z n (t).