Scaling limits of multitype Bienaym\'e trees

TL;DR

This paper investigates the scaling limits of critical multitype Bienaymé trees under various conditioning schemes, demonstrating convergence to the Brownian Continuum Random Tree and providing tail bounds for tree height.

Contribution

It establishes the convergence of multitype Bienaymé trees to the Brownian CRT under specific conditions and introduces new structural analysis tools for these trees.

Findings

Convergence to Brownian CRT under linear and subset conditioning.

Strong tail bounds for the height of multitype trees.

Development of a flattening operation and degree estimates for analysis.

Abstract

We consider critical multitype Bienaym\'e trees that are either irreducible or possess a critical irreducible component with attached subcritical components. These trees are studied under two distinct conditioning frameworks: first, conditioning on the value of a linear combination of the numbers of vertices of given types; and second, conditioning on the precise number of vertices belonging to a selected subset of types. We prove that, under a finite exponential moment condition, the scaling limit as the tree size tends to infinity is given by the Brownian Continuum Random Tree. Additionally, we establish strong non-asymptotic tail bounds for the height of such trees. Our main tools include a flattening operation applied to multitype trees and sharp estimates regarding the structure of monotype trees with a given sequence of degrees.