## Electronic Journal of Probability

### Sublinearity of the number of semi-infinite branches for geometric random trees

David Coupier

#### Abstract

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.

#### Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 37, 33 pp.

Dates
Accepted: 8 October 2017
First available in Project Euclid: 9 May 2018

https://projecteuclid.org/euclid.ejp/1525852814

Digital Object Identifier
doi:10.1214/17-EJP115

Mathematical Reviews number (MathSciNet)
MR3806405

Zentralblatt MATH identifier
1390.60048

#### Citation

Coupier, David. Sublinearity of the number of semi-infinite branches for geometric random trees. Electron. J. Probab. 23 (2018), paper no. 37, 33 pp. doi:10.1214/17-EJP115. https://projecteuclid.org/euclid.ejp/1525852814

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