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Scaling limits of random trees.

In this thesis we investigate the asymptotic behavior of large random Galton-Watson (GW) -trees. It is found that in the space of compact real trees equipped with Gromov-Hausdorff metric a limit in distribution can be defined. This limit is a real tree coded by the normalized excursion of Brownian motion. First, Itōs excursion measure of the Brownian motion is intro- duced. Using the Markov property of Brownian motion the law of the normalized Brownian excursion is specified. With this result and the methods of Le Gall and Miermont, [14], a limit theorem of the excursion of a simple random walk towards the normalized excursion is proven. An excursion of a simple random walk coincides with the distribution of a GW-tree’s contour process. It is shown that similarly arbitrary continuous functions of the unit interval yield a real tree using a rerooting isometry. Random trees can then be seen as random variables taking values in the space of real compact trees endowed with the pointed Gromov-Hausdorff topology. With respect to this metric, the mapping from the space of excursions to the space of compact real trees turns out to be continuous. Using the continuous mapping theorem and the limit theorem of excursions it follows that the sequence of rescaled GW-trees converges in distribution towards Aldous’ Continuum Random Tree (CRT, [1]).

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Year: 2019

Author : Tobias Schröder

Prof. Dr. Anna Wienhard