Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A degree-biased cutting process for random recursive trees

Laura Eslava, Sergio I. López, Marco L. Ortiz

TL;DR

The paper addresses how a degree-biased vertex-cutting procedure on random recursive trees erodes the tree and how many cuts are needed to erase it. It establishes the splitting property, derives the mass distribution per cut, and obtains a recursive formula for $K_n$, the number of cuts to erase a size-$n$ tree, while showing that $K_n$ is stochastically dominated by a barrier-walk jump process $J_n$ whose asymptotics are tractable. The analysis yields precise first-order and limiting behavior: ${f E}[J_n]\sim 2n/(\\ln n)^2$, and suitably normalized $J_n$ converges to a spectrally negative stable law, with a stronger 3–5 sentence convergence description for $J_n$ after centering. The work also connects the cutting procedure to a coalescent process, deriving explicit first-coalescence rates for the degree-biased coalescent and highlighting why these rates do not form a $\,\Lambda$-coalescent, thereby enriching the understanding of dynamic random trees and their associated coalescent structures.

Abstract

We investigate a degree-biased cutting process on random recursive trees, where each vertex is deleted with probability proportional to its degree. We establish the splitting property and derive the explicit distribution of the number of vertices deleted in each cut. This leads to a recursive formula for Kn, the number of cuts needed to erase a random recursive tree with n vertices. Furthermore, we show that Kn is stochastically dominated by Jn, the number of jumps made by a related walk with a barrier. We prove that Jn converges in distribution to a random variable with a spectrally negative stable distribution. Finally, we examine connections between this cutting procedure and a coalescing process on the set of n elements.

A degree-biased cutting process for random recursive trees

TL;DR

The paper addresses how a degree-biased vertex-cutting procedure on random recursive trees erodes the tree and how many cuts are needed to erase it. It establishes the splitting property, derives the mass distribution per cut, and obtains a recursive formula for , the number of cuts to erase a size- tree, while showing that is stochastically dominated by a barrier-walk jump process whose asymptotics are tractable. The analysis yields precise first-order and limiting behavior: , and suitably normalized converges to a spectrally negative stable law, with a stronger 3–5 sentence convergence description for after centering. The work also connects the cutting procedure to a coalescent process, deriving explicit first-coalescence rates for the degree-biased coalescent and highlighting why these rates do not form a -coalescent, thereby enriching the understanding of dynamic random trees and their associated coalescent structures.

Abstract

We investigate a degree-biased cutting process on random recursive trees, where each vertex is deleted with probability proportional to its degree. We establish the splitting property and derive the explicit distribution of the number of vertices deleted in each cut. This leads to a recursive formula for Kn, the number of cuts needed to erase a random recursive tree with n vertices. Furthermore, we show that Kn is stochastically dominated by Jn, the number of jumps made by a related walk with a barrier. We prove that Jn converges in distribution to a random variable with a spectrally negative stable distribution. Finally, we examine connections between this cutting procedure and a coalescing process on the set of n elements.
Paper Structure (9 sections, 8 theorems, 51 equations)

This paper contains 9 sections, 8 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Properties of the degree-biased cutting process
  3. Main results
  4. Jumps for random walks with a barrier
  5. Uniform random cuts
  6. Analysis of the degree-biased cutting process
  7. Harmonic-biased random walk with a barrier
  8. A related coalescent process
  9. Degree-biased version

Key Result

Proposition 1.2

Let $n\ge 2$ and let $T_n$ be a random recursive tree. Let $T_n^{root}$ be the tree obtained after a degree-biased cut. For $\ell\in[n-2]$, conditionally given that $\vert T_n^{root}\vert=\ell$, $\Phi (T_n^{root})$ has the distribution of a random recursive tree of size $\ell$.

Theorems & Definitions (17)

  • Definition 1.1: Degree-biased cut
  • Proposition 1.2: Splitting property
  • Proposition 1.3: Size of a cut
  • Definition 1.4: Degree-biased process
  • Theorem 1.5
  • Theorem 1.6
  • Proposition 2.2: Theorem 1.4 in Iksanov2008
  • Lemma 3.1
  • proof
  • proof : Proofs of Propositions \ref{['prop: split']} and \ref{['prop: size']}
  • ...and 7 more