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2018 Sublinearity of the number of semi-infinite branches for geometric random trees
David Coupier
Electron. J. Probab. 23: 1-33 (2018). DOI: 10.1214/17-EJP115


The present paper addresses the following question: for a geometric random tree in $\mathbb{R} ^{2}$, how many semi-infinite branches cross the circle $\mathcal{C} _{r}$ centered at the origin and with a large radius $r$? We develop a method ensuring that the expectation of the number $\chi _{r}$ of these semi-infinite branches is $o(r)$. The result follows from the fact that, far from the origin, the distribution of the tree is close to that of an appropriate directed forest which lacks bi-infinite paths. In order to illustrate its robustness, the method is applied to three different models: the Radial Poisson Tree (RPT), the Euclidean First-Passage Percolation (FPP) Tree and the Directed Last-Passage Percolation (LPP) Tree. Moreover, using a coalescence time estimate for the directed forest approximating the RPT, we show that for the RPT $\chi _{r}$ is $o(r^{1-\eta })$, for any $0<\eta <1/4$, almost surely and in expectation.


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David Coupier. "Sublinearity of the number of semi-infinite branches for geometric random trees." Electron. J. Probab. 23 1 - 33, 2018.


Received: 3 January 2017; Accepted: 8 October 2017; Published: 2018
First available in Project Euclid: 9 May 2018

zbMATH: 1390.60048
MathSciNet: MR3806405
Digital Object Identifier: 10.1214/17-EJP115

Primary: 60D05


Vol.23 • 2018
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