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 $2^{-n} Z_n(T_n + n \tau)$, $\tau\in\bb R$, converges to the Gompertz curve $\exp (- (\ln 2) e^{-\beta \tau})$. We also prove a central limit theorem for the martingale associated to $Z_n(t)$.
- Publication:
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arXiv e-prints
- Pub Date:
- July 2008
- DOI:
- 10.48550/arXiv.0807.1750
- arXiv:
- arXiv:0807.1750
- Bibcode:
- 2008arXiv0807.1750L
- Keywords:
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- Quantitative Biology - Cell Behavior;
- Mathematical Physics;
- Quantitative Biology - Quantitative Methods;
- Statistics - Other Statistics
- E-Print:
- 13 pages