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

Abstract : The present paper addresses the following question: for a geometric random tree in R-2, how many semi-infinite branches cross the circle C-r centered at the origin and with a large radius r? We develop a method ensuring that the expectation of the number X-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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Submitted on : Monday, May 17, 2021 - 1:32:27 PM
Last modification on : Monday, November 22, 2021 - 4:31:33 PM

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David Coupier. Sublinearity of the number of semi-infinite branches for geometric random trees. Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2018, 23, ⟨10.1214/17-EJP115⟩. ⟨hal-03227549⟩

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