![]() ![]() This effectively means that reproduction can be modeled as asexual. arXiv:2201.01899v1 math. Start with a root vertex, and attach edges from to D 0 first-generation vertices. In the classical family surname GaltonWatson process described above, only men need to be considered, since only males transmit their family name to descendants. Keywords: bootstrap percolation branching. This yields a multi-dimensional extension of a limit theorem due to Janson for the number of random cuts needed to isolate the root in Galton-Watson trees conditioned by their sizes, and also generalizes a recent result obtained in the special case of Cayley trees. The Galton-Watson tree with initial degree distribution D 0 and general degree distribution D g is constructed as follows. of GaltonWatson trees with critical probability bounded from above by Cre b r1 for some constant Cr > 0. Our main result shows that after a proper rescaling, the cut-tree of a critical Galton-Watson tree with finite variance and conditioned to have size n, converges as n to a Brownian continuum random tree (CRT) in the weak sense induced by the Gromov-Prokhorov topology. Roughly speaking, the cut-tree then describes the genealogy of connected components appearing in this edge-deletion process. GaltonWatson Trees Semantic Scholar We recall a few elementary properties of supercritical GaltonWatson trees, and introduce the notion of size-biased trees. Copy AbstractĬonsider the edge-deletion process in which the edges of some finite tree T are removed one after the other in the uniform random order. The cut-tree of large galton-watson trees and the brownian crt. Percolation On Galton-Watson Trees Abstract We consider both Bernoulli and invasion percolation on Galton-Watson trees. ![]()
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