Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Existence of critical tiltings and local limits of general size-conditioned Bienaymé-Galton-Watson multitype trees

Rémy Poudevigne, Paul Thévenin

TL;DR

This work develops a comprehensive framework for size-conditioned multitype BGW trees by introducing exponential tiltings and a convex-geometric apparatus centered on the function $\chi$ and the set $\mathcal{C}_{image}$. It proves that under regularity hypotheses (entire, finite, nondegenerate, irreducible, aperiodic), there exists a nonempty, smooth manifold of critical tiltings $\mathcal{M}_{crit}$ whose boundary corresponds to $\partial\mathcal{C}_{image}$ via $\chi$, and that these tiltings govern the asymptotic composition of types in large conditioned trees. A key outcome is a dichotomy: conditioned noncritical BGW trees can be analyzed through equivalent critical tiltings, enabling local limit results to be established and framed in terms of multitype Kesten trees with spines and grafted subtrees. The paper also clarifies the geometry of asymptotic directions, proves their accessibility properties, and provides counterexamples showing the necessity of entireness, enriching the understanding of when such equivalences hold and how general conditioning shapes local limits.

Abstract

We are interested in the structure of multitype Bienaymé-Galton-Watson (BGW) trees conditioned on integer linear combinations of the numbers of vertices of given types. We show that, under regularity assumptions on the offspring distributions, it is always possible to find a critical BGW tree having the same conditional distribution. This allows us to prove the existence of local limits for noncritical BGW trees, under a large variety of conditionings. Our proof is based on geometric considerations on the set of the so-called exponential tiltings of a family of offspring distributions.

Existence of critical tiltings and local limits of general size-conditioned Bienaymé-Galton-Watson multitype trees

TL;DR

This work develops a comprehensive framework for size-conditioned multitype BGW trees by introducing exponential tiltings and a convex-geometric apparatus centered on the function and the set . It proves that under regularity hypotheses (entire, finite, nondegenerate, irreducible, aperiodic), there exists a nonempty, smooth manifold of critical tiltings whose boundary corresponds to via , and that these tiltings govern the asymptotic composition of types in large conditioned trees. A key outcome is a dichotomy: conditioned noncritical BGW trees can be analyzed through equivalent critical tiltings, enabling local limit results to be established and framed in terms of multitype Kesten trees with spines and grafted subtrees. The paper also clarifies the geometry of asymptotic directions, proves their accessibility properties, and provides counterexamples showing the necessity of entireness, enriching the understanding of when such equivalences hold and how general conditioning shapes local limits.

Abstract

We are interested in the structure of multitype Bienaymé-Galton-Watson (BGW) trees conditioned on integer linear combinations of the numbers of vertices of given types. We show that, under regularity assumptions on the offspring distributions, it is always possible to find a critical BGW tree having the same conditional distribution. This allows us to prove the existence of local limits for noncritical BGW trees, under a large variety of conditionings. Our proof is based on geometric considerations on the set of the so-called exponential tiltings of a family of offspring distributions.

Paper Structure

This paper contains 28 sections, 34 theorems, 136 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Outline of the paper
  4. Background on random trees
  5. Plane trees
  6. Multitype plane trees
  7. Multitype BGW trees
  8. Size-conditioned BGW trees
  9. Local limit of trees
  10. Statement of the results
  11. Exponential tiltings and asymptotic directions
  12. The function $\chi$ and $\Gamma$-equivalence
  13. Main results
  14. Properties of the set $\mathcal{M}_{crit}$ of critical tiltings
  15. Critical tiltings as maximizers
  16. ...and 13 more sections

Key Result

Proposition 3.5

Fix $i \in [K],\ {\boldsymbol{\theta}},{\boldsymbol{\theta}}' \in \mathbb{R}^K$ such that ${\boldsymbol{\theta}} \sim_\Gamma {\boldsymbol{\theta}}'$, and $\mathbf{g} \in \mathbb{Z}^\ell$ such that $\mathbb{P}(\Gamma \ \mathbf{N}(\mathcal{T}^{{\boldsymbol{\theta}},(i)})=\mathbf{g})>0$. Then, the foll

Figures (1)

  • Figure 1: In dashed blue, the set $\mathcal{C}_{image}$ for the projection $\boldsymbol{\mu}$ with generating functions $\phi^{(1)}(x_1,x_2) = \frac{1}{3}\left( x_1x_2^2 + x_1x_2 + x_2 \right)$ and $\phi^{(2)}(x_1,x_2) = \frac{1}{3}\left( x_1x_2+x_2+1 \right)$. Non The boundary of $\mathcal{C}_{image}$, in blue, has parametrization: Oui$((2s-2\ln(1-2e^s)+\ln(1-e^s)-\ln(3)-\ln(1+e^s)),(\ln(1-2e^{2s}(1+e^s))-\ln(3)-2s))$, for $s\in(-\infty,-\ln(2))$. Peut-être

Theorems & Definitions (80)

  • Definition 2.1: Size-conditioned tree
  • Definition 2.2
  • Definition 3.1: Exponential tiltings
  • Definition 3.2: Asymptotic direction
  • Definition 3.3
  • Definition 3.4: $\Gamma$-equivalent tiltings
  • Proposition 3.5
  • proof
  • Definition 3.6: Asymptotic cone
  • Theorem 3.7
  • ...and 70 more