[Paper Review] Scaling limit of multitype Galton-Watson trees with infinitely many types

Loïc de Raphélis|arXiv (Cornell University)|May 15, 2014

Stochastic processes and statistical mechanics16 references16 citations

TL;DR

This paper establishes the scaling limit of multitype Galton–Watson forests with a countably infinite number of types by introducing a novel class of 2-type Galton–Watson trees with edge lengths—'leafed' trees where one type acts as sterile leaves. Under mild moment and ergodicity conditions, the weighted height function of such forests converges in distribution to reflected Brownian motion, extending Miermont's result for finite-type forests to the infinite-type case via a tree-reduction technique.

ABSTRACT

We introduce a certain class of 2-type Galton-Watson trees with edge lengths. We prove that, after an adequate rescaling, the weighted height function of a forest of such trees converges in law to the reflected Brownian motion. We then use this to deduce under mild conditions an invariance principle for multitype Galton--Watson trees with a countable number of types, thus extending a result of G. Miermont on multitype Galton--Watson trees with finitely many types.

Motivation & Objective

• To extend G. Miermont's invariance principle for critical multitype Galton–Watson forests with finitely many types to the case of countably infinite types.
• To establish a scaling limit for the weighted height function of such forests under mild moment and ergodicity conditions.
• To introduce and analyze 'leafed Galton–Watson trees with edge lengths' as a technical tool to reduce the infinite-type problem to a tractable 2-type framework.
• To prove convergence in law of the height function to reflected Brownian motion using a tree-reduction method inspired by Miermont's approach.
• To provide sufficient conditions—specifically geometric ergodicity and moment bounds—under which the scaling limit holds.

Proposed method

• Define a 2-type Galton–Watson process with edge lengths, where type 1 vertices reproduce and type 0 vertices are sterile (acting as artificial leaves), forming 'leafed Galton–Watson trees'.
• Construct a forest of i.i.d. such trees and code them using Neveu's notation on the Ulam–Harris tree to handle planarity and ancestry.
• Introduce a Markov chain on the types induced by the ancestral lineages, and impose a geometric ergodicity condition (H_R^{alt}) via a Lyapunov function V and drift condition.
• Use a change of measure via the left eigenvector (b_x) to relate expectations under the original measure to a transformed measure where the ancestral process is a Markov chain.
• Apply moment bounds and exponential moment estimates from Markov chain theory (specifically Theorem 15.2.6 in [17]) to control the tail behavior of hitting times and height function moments.
• Reduce the height function of a general multitype tree to that of a leafed tree via a tree-reduction method, enabling the application of the 2-type limit theorem to the infinite-type case.

Experimental results

Research questions

• RQ1 Can the invariance principle for multitype Galton–Watson forests be extended from finitely many types to a countably infinite number of types?
• RQ2 Under what conditions does the weighted height function of a multitype Galton–Watson forest with infinitely many types converge to reflected Brownian motion?
• RQ3 How can the height process of a general multitype tree be related to that of a 2-type leafed Galton–Watson tree with edge lengths?
• RQ4 What moment and ergodicity conditions ensure the tightness and convergence of the rescaled height function in the infinite-type setting?
• RQ5 Is geometric ergodicity of the ancestral type process sufficient to guarantee the required moment bounds for the scaling limit?

Key findings

• Under the geometric ergodicity condition (H_R^{alt}), the weighted height function of a leafed Galton–Watson forest with edge lengths converges in distribution to reflected Brownian motion after appropriate rescaling.
• The condition (H_R^{alt}) implies the stronger condition (H_R^{x_0}) for any initial type x_0, ensuring the necessary moment bounds on the height and return time to x_0.
• The finiteness of the second moment of the hitting time τ_{x_0} is guaranteed by the exponential moment condition derived from geometric ergodicity, which ensures tightness of the height process.
• The tree-reduction method successfully links the height process of a general multitype Galton–Watson tree to that of a leafed 2-type tree, enabling the extension of the limit theorem.
• The result generalizes Miermont's invariance principle from finite to countably infinite type spaces, under mild moment and ergodicity assumptions.
• The convergence holds even when the type space is infinite, provided the ancestral process is geometrically ergodic and the edge lengths satisfy integrability conditions.