Record process on the Continuum Random Tree

Abstract

By considering a continuous pruning procedure on Aldous's Brownian tree, we construct a random variable $\Theta$ which is distributed, conditionally given the tree, according to the probability law introduced by Janson as the limit distribution of the number of cuts needed to isolate the root in a critical Galton-Watson tree. We also prove that this random variable can be obtained as the a.s. limit of the number of cuts needed to cut down the subtree of the continuum tree spanned by $n$ leaves.

References in corpus (5)

• A note on Gromov-Hausdorff-Prokhorov distance between (locally) compact measure spaces
• Cutting down trees with a Markov chainsaw
• Growth of Levy trees
• The coding of compact real trees by real valued functions
• Exit times for an increasing Lévy tree-valued process

Cited by in corpus (11)

• Cutting down trees with a Markov chainsaw
• Convergence of bi-measure R-tree and the pruning process
• Multiple isolation of nodes in recursive trees
• Exit times for an increasing Lévy tree-valued process
• Pruning of CRT-sub-trees
• Reversing the cut tree of the Brownian continuum random tree
• A construction of a $\beta$-coalescent via the pruning of Binary Trees
• Gromov-Hausdorff-Prokhorov convergence of vertex cut-trees of n-leaf Galton-Watson trees
• Cutting down $\mathbf{p}$-trees and inhomogeneous continuum random trees
• $k$-cut model for the Brownian Continuum Random Tree
• Fluctuations for the number of records on subtrees of the Continuum Random Tree