Multitype Lévy trees as scaling limits of multitype Bienaymé-Galton-Watson trees

TL;DR

The paper proves an invariance principle for sequences of large multitype Bienaymé-Galton-Watson trees conditioned to be large, showing convergence to a multitype Lévy tree in the Gromov-Hausdorff-Prohorov topology. The approach combines discrete encodings via $Z^d$-valued walks, a robust glued-decoration framework (building on Sénizergues) with the Ulam tree, and continuum constructions from spectrally positive multitype Lévy processes. The main contribution is a general convergence theorem for glued decorations that yields the conditioned MBGW-to-multitype Lévy tree limit, together with detailed analysis of multidimensional first hitting times, excursions, and height processes. This work extends the continuum random tree framework to the multitype setting, enabling potential applications to scaling limits of multitype random graphs and related combinatorial structures. The results provide a rigorous, modular pathway from discrete multitype branching structures to a unified multitype CRT-like object driven by spaLf exponentials and Lévy fields, with implications for understanding the geometry of multitype random graphs at criticality.

Abstract

We establish sufficient mild conditions for a sequence of multitype Bienaymé-Galton-Watson trees, conditioned in some sense to be large, to converge to a limiting compact metric space which we call a \emph{multitype Lévy tree}. More precisely, we condition on the size of the maximal subtree of vertices of the same type generated by the root to be large. Although under a different conditioning, our result can be seen as a generalization to the multitype setting of the continuum random trees defined by Aldous, Duquesne and Le Gall in [Ald91a,Ald91b,Ald93,DLG02]. Our main result is an invariance principle for the convergence of such trees, by gluing single-type Lévy trees together in a method determined by the limiting spectrally positive additive Lévy field, as constructed by Chaumont and Marolleau [CM21]. Our approach is a particular case of a more general result about the convergence in the Gromov-Hausdorff-Prohorov topology, of compact marked metric spaces equipped with vector-valued measures, and then glued via an iterative operation. To analyze the gluing operation, we extend the techniques developed by Sénizergues [Sen19,Sen22] to the multitype setting. While the single-type case exhibits a more homogeneous structure with simpler dependency patterns, the multitype case introduces interactions between different types, leading to a more intricate dependency structure where functionals must account for type-specific behaviors and inter-type relationships.

Figures (10)

• Figure 1: We show a simulation of a multitype tree with $d = 3$ and 32000 vertices.
• Figure 2: We show a multitype tree with $d = 3$, where type one is represented as black circles, type two as red squares, and type three as blue pentagons. On the left, the tree together with the depth-first order of the subtree ${\tt t}^1_1$ (the first subtree type one) is depicted. On the right, we show the corresponding reduced tree together with its breadth-first order.
• Figure 3: We depict the paths $(X^{1}_{{\tt t}^1_1},X^{2}_{{\tt t}^1_1},X^{3}_{{\tt t}^1_1})$ of the subtree ${\tt t}^1_1$.
• Figure 4: The reduced subtree together with the labeling via the Ulam-Harris tree. Recall that type $1$ individuals are black, type $2$ are red, and type $3$ are blue.
• Figure 5: The type $j$ subtrees at reduced height $0$, $1$, 2 and 3. Since the red square vertices are type $2$, the corresponding subtrees are assigned indexes $2,5,8,11,\dotsm\in {\mathbb{U}}$. Those type 2 subtrees are also labeled ${\tt t}'_{2,(1)}, {\tt t}'_{2,(2)}, {\tt t}'_{2,(3)}$. The blue vertices are type $3$ and so the corresponding subtrees are indexed by $3,6,9,\dotsm$. The type $2$ subtree in the right (in red) is labeled ${\tt t}_2'$ because it is the largest type $2$ subtree at reduced height $1$.

Theorems & Definitions (67)

• Theorem 1.1: Informal statement of Theorem \ref{['thm:MAIN']}
• Definition 2.1: Decoration, glued subdecorated space and glued decoration
• Proposition 2.3
• Theorem 2.4
• Remark 2.5
• Remark 2.6
• Definition 2.7: Bienaymé-Galton-Watson Trees
• Proposition 2.8: AN.72
• Remark 2.11
• Remark 2.12