[Paper Review] Local convergence of critical random trees and continuous-state branching processes

Xin He|arXiv (Cornell University)|Mar 3, 2015

Stochastic processes and statistical mechanics10 references3 citations

TL;DR

This paper establishes a general framework for the local convergence of critical random trees—Galton-Watson trees, Lévy trees, and continuous-state branching processes—under various conditionings, showing they converge to immortal or condensation trees. Under a monotonicity condition on functionals like height, width, or total mass, conditioned critical trees converge locally to size-biased trees with infinite spine (immortal trees), with a general ratio limit property derived for functionals satisfying this condition.

ABSTRACT

We study the local convergence of critical Galton-Watson trees and Levy trees under various conditionings. Assuming a very general monotonicity property on the functional of random trees, we show that random trees conditioned to have large functional values always converge locally to immortal trees. We also derive a very general ratio limit property for functionals of random trees satisfying the monotonicity property. Then we move on to study the local convergence of critical continuous-state branching processes, and prove a similar result. Finally we give a definition of continuum condensation trees, which should be the correct local limits for certain subcritical Levy trees under suitable conditionings.

Motivation & Objective

• To establish a general theory for local convergence of critical Galton-Watson trees under diverse conditionings, such as large height, width, or maximal degree.
• To extend the local convergence framework to Lévy trees and continuous-state branching processes (CB processes), proving convergence to continuum immortal trees under the same monotonicity condition.
• To introduce and define continuum condensation trees as candidate local limits for subcritical Lévy trees under large maximal degree or total mass conditioning.
• To formulate and motivate two conjectures on the local convergence of subcritical Lévy trees to continuum condensation trees under large total mass and large maximal degree.
• To identify an open problem on the local limit of subcritical trees under large width conditioning, which remains unresolved.

Proposed method

• Introduce a general monotonicity property on functionals of random trees, ensuring that conditioning on large functional values leads to local convergence to immortal trees.
• Use a novel method based on criticality and spine decomposition, differing from prior frameworks, to prove local convergence for Galton-Watson trees (Theorem 2.1) and Lévy trees (Theorem 4.1).
• Derive a general ratio limit property (Theorem 2.7 and Theorem 4.5) for functionals satisfying the monotonicity condition, applicable to width, total mass, and maximal degree.
• Adapt the proof techniques to continuous-state branching processes (CB processes), showing convergence to CBI processes with immigration under the same monotonicity condition (Theorem 4.7).
• Define continuum condensation trees via excursion measures and excursion local time, inspired by condensation phenomena in subcritical branching processes.
• Use excursion theory and the canonical process under excursion measure $\mathbf{N}$ to characterize the limiting behavior of height processes under large jump or large mass conditioning.

Experimental results

Research questions

• RQ1 Under what general conditions do critical Galton-Watson trees converge locally to immortal trees when conditioned on large values of a functional?
• RQ2 Can the same convergence result be extended to Lévy trees and continuous-state branching processes under the same monotonicity condition?
• RQ3 What is the correct limiting object for subcritical Lévy trees under conditioning on large maximal degree or large total mass?
• RQ4 Is the local limit of a subcritical Galton-Watson tree under large width conditioning an immortal tree or a condensation tree?
• RQ5 Do continuum condensation trees arise as weak limits of Lévy trees under large maximal degree or large total mass conditioning?

Key findings

• Critical Galton-Watson trees conditioned on large functional values (e.g., height, width, maximal degree) converge locally to an immortal tree under a general monotonicity condition on the functional (Theorem 2.1).
• A general ratio limit property is established for functionals of Galton-Watson trees satisfying the monotonicity condition, with explicit results for width (Proposition 2.8).
• Critical Lévy trees conditioned on large functional values (e.g., width, total mass, maximal degree) converge locally to a continuum immortal tree under the same monotonicity condition (Theorem 4.1).
• A general ratio limit property is derived for Lévy trees under the same condition, extending the result to continuous-state processes (Theorem 4.5).
• Critical continuous-state branching processes (CB processes) conditioned on large functional values converge locally to CBI processes with immigration under the monotonicity condition (Theorem 4.7).
• Two conjectures are proposed: Lévy trees under large maximal degree or large total mass conditioning converge weakly to continuum condensation trees (Conjectures 5.1 and 5.2), though proofs remain open.