Random real trees

By Jean-François Le Gall. 2006

91 citations recorded by the source corpus.

We survey recent developments about random real trees, whose prototype is the Continuum Random Tree (CRT) introduced by Aldous in 1991. We briefly explain the formalism of real trees, which yields a neat presentation of the theory and in particular of the relations between discrete Galton-Watson trees and continuous random trees. We then discuss the particular class of self-similar random real trees called stable trees, which generalize the CRT. We review several important results concerning stable trees, including their branching property, which is analogous to the well-known property of Galton-Watson trees, and the calculation of their fractal dimension. We then consider spatial trees, which combine the genealogical structure of a real tree with spatial displacements, and we explain their connections with superprocesses. In the last section, we deal with a particular conditioning problem for spatial trees, which is closely related to asymptotics for random planar quadrangulations.

Subfields: Management Science and Operations Research, Mathematical Physics, Signal Processing

View Paper: https://www.semanticscholar.org/paper/132f728853d5a1d9a527dee7c86efc78a8664cd0

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In Management Science and Operations Research

In Mathematical Physics

In Signal Processing

Citations in the corpus

The local weak limit of the minimum spanning tree of the complete graph (2013, background): “Then d̂ induces a distance on T , still denoted by d̂, and it is standard (see, for example, [44]) that (T , d̂) is a compact R-tree.”
A continuum-tree-valued Markov process (2009, background): “The results of this section are mainly extracted from [17], except for the next subsection which is extracted from [22].”
Scaling limits of Markov branching trees and Galton-Watson trees conditioned on the number of vertices with out-degree in a given set (2011, background): “Subsequent work on scaling limits of random trees proceeded largely in two directions, one focusing on conditioned Galton–Watson trees and other trees with nice encodings by continuous functions (see [8]”
Cutting down trees with a Markov chainsaw (2011, methodology): “with root ˆand distance metric d, let be its mass measure, and let ‘be the length measure on the skeleton of T. We will use the following fundamental result heavily. Theorem 5.2 (Aldous [6], Le Gall [31]). It holds that (Tn;dn;n;ˆn) !d (T;d;;ˆ) as n!1, where convergence is in the 1-pointed Gromov–Ha”
Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation (2011, background): “ee is not an ordinary tree; it is a compact metric space.) This has been extended to the case σ2 = ∞ when π is in the domain of attraction of a stable distribution, see e.g. Duquesne [34] and Le Gall [80, 81]; the limit is now a diﬀerent random metric space called a stable tree. Problem 20.11. Is th”
A view from infinity of the uniform infinite planar quadrangulation (2012, background): “It is a standard result [25] that the distribution of a uniformly chosen planar tree Tn with n edges is the same as the distribution of a critical Galton-Watson tree with geometric offspring distribution conditioned on the total progeny to be n+1.”
The continuum limit of critical random graphs (2009, unknown): “Theorem 4 (Aldous [6], Le Gall [34]).”