On the non-Gaussian fluctuations of the giant cluster for percolation on random recursive trees. (English) Zbl 1292.60095

Bertoin, Jean

Electron. J. Probab. 19, Paper No. 24, 15 p. (2014).

Summary: We consider a Bernoulli bond percolation on a random recursive tree of size \(n\gg 1\), with supercritical parameter \(p_n=1-c/\ln n\) for some \(c>0\) fixed. It is known that with high probability, there exists then a unique giant cluster of size \(G_n\sim e^{-c}n\), and it follows from a recent result of J. Schweinsberg [Electron. J. Probab. 17, Paper No. 91, 50 p. (2012; Zbl 1284.92072)] that \(G_n\) has non-Gaussian fluctuations. We provide an explanation of this by analyzing the effect of percolation on different phases of the growth of recursive trees. This alternative approach may be useful for studying percolation on other classes of trees, such as for instance regular trees.

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
05C05	Trees
60F05	Central limit and other weak theorems
60J80	Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords:

random recursive tree; giant cluster; fluctuations; super-critical percolation

Citations:

Zbl 1284.92072