Convergence to the Brownian CRT for critical branching Markov processe

TL;DR

This paper proves that the genealogical trees of a broad class of critical branching Markov processes with finite variance converge to the Brownian continuum random tree under rescaling, revealing a universal scaling limit.

Contribution

It establishes a universal invariance principle for critical branching processes with non-local mechanisms, showing convergence to the Brownian CRT.

Findings

Genealogical trees converge to the Brownian CRT

Convergence holds in Gromov-Hausdorff-weak topology

Universal scaling limit for critical finite variance branching processes

Abstract

We prove an invariance principle for a general class of continuous time critical branching processes with finite variance (non-local) branching mechanism. We show that the genealogical trees, viewed as random compact metric measure spaces, converge under rescaling to the Brownian continuum random tree in the Gromov-Hausdorff-weak topology, establishing a universal scaling limit for critical finite variance branching processes.