Abstract
© 2015, University of Washington. All rights reserved. In this note, we provide a new characterization of Aldous’ Brownian continuum random tree as the unique fixed point of a certain natural operation on continuum trees (which gives rise to a recursive distributional equation). We also show that this fixed point is attractive.
Highlights
The Brownian continuum random tree (BCRT), which was introduced and first studied by Aldous [3, 4, 5], is the prototypical example of a random R-tree/continuum random tree
Its importance derives from the fact that it is the scaling limit of a large class of discrete trees including: all critical Galton–Watson trees with finite offspring variance [3, 5], unordered binary trees [17], uniform unordered trees [14], uniform unlabelled unrooted trees [24], critical multitype Galton–Watson trees [18] and random trees with a prescribed degree sequence satisfying certain conditions [8]
We prove that the BCRT is the unique fixed point of an appropriate operator, and that this fixed point is attractive for a certain natural class of measures on continuum trees
Summary
The Brownian continuum random tree (BCRT), which was introduced and first studied by Aldous [3, 4, 5], is the prototypical example of a random R-tree/continuum random tree. Banach’s fixed point theorem tells us that there exists a fixed point and that Mn → M as n → ∞ in the sense of that metric This is straightforward in principle, but usually the recursive equation for Mn does not have precisely the same form as the limiting operator. Aldous [6] proved that the BCRT is a fixed point for a natural operation on continuum trees With these two facts in mind, it is natural to ask if a contraction method can be established for random trees.
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