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

2015/01/20 by David Coupier, Coupier, David
Mathematics · #60D05 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1501.04804

openalex publication_date 2015/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The present paper addresses the following question: for a geometric random\ntree in R², how many semi-infinite branches cross the circle CC_r\ncentered at the origin and with a large radius r? We develop a method\nensuring that the expectation of the number \χ_r of these semi-infinite\nbranches is o(r). The result follows from the fact that, far from the origin,\nthe distribution of the tree is close to that of an appropriate directed forest\nwhich lacks bi-infinite paths. In order to illustrate its robustness, the\nmethod is applied to three different models: the Radial Poisson Tree (RPT), the\nEuclidean First-Passage Percolation (FPP) Tree and the Directed Last-Passage\nPercolation (LPP) Tree. Moreover, using a coalescence time estimate for the\ndirected forest approximating the RPT, we show that for the RPT \χ_r is\no(r^1-\η), for any 0<\η<1/4, almost surely and in expectation.\n

Citations

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• Limiting shape for directed percolation models

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