Multitype L\'evy trees as scaling limits of multitype Bienaym\'e-Galton-Watson trees

TL;DR

This paper proves that multitype Bienaymé-Galton-Watson trees, under certain conditions, converge to multitype Lévy trees, extending the theory of continuum random trees to a more complex multitype setting.

Contribution

It introduces a new invariance principle for multitype trees, generalizing single-type results by gluing Lévy trees according to a spectrally positive Lévy field.

Findings

Established convergence conditions for multitype trees to Lévy trees.

Extended gluing techniques to the multitype setting.

Analyzed complex inter-type dependencies in multitype structures.

Abstract

We establish sufficient mild conditions for a sequence of multitype Bienaym\'e-Galton-Watson trees, conditioned in some sense to be large, to converge to a limiting compact metric space which we call a \emph{multitype L\'{e}vy tree}. More precisely, we condition on the size of the maximal subtree of vertices of the same type joined by the root to be large. While we employ 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 L\'{e}vy trees together in a method determined by the limiting spectrally positive additive L\'{e}vy field, as constructed by Chaumont and Marolleau [CM21]. Our approach is an improvement of a result about the convergence in the Gromov-Hausdorff-Prohorov topology, of compact marked metric spaces equipped with vector-valued measures, which are then glued via an iterative operation. To analyze the gluing operation, we extend the techniques developed by S\'enizergues [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.