Conditioning (sub)critical L{\'e}vy trees by their maximal degree: Decomposition and local limit

TL;DR

This paper analyzes the structure and distribution of maximal degrees in (sub)critical Lévy trees, providing decomposition results, conditioned distributions, and local limits, extending understanding of these trees beyond classical Grey condition assumptions.

Contribution

It introduces a detailed genealogical and distributional analysis of large nodes in Lévy trees, including a Poissonian decomposition and conditioned structures, without relying on the Grey condition.

Findings

Maximal degree is realized by a unique node with exponential height distribution.

Conditioned Lévy trees can be constructed by grafting Lévy forests onto size-biased Lévy trees.

Large maximal degree conditioned trees converge locally to an immortal or condensation tree depending on criticality.

Abstract

We study the maximal degree of (sub)critical L{\'e}vy trees which arise as the scaling limits of Bienaym{\'e}-Galton-Watson trees. We determine the genealogical structure of large nodes and establish a Poissonian decomposition of the tree along those nodes. Furthermore, we make sense of the distribution of the L{\'e}vy tree conditioned to have a fixed maximal degree. In the case where the L{\'e}vy measure is diffuse, we show that the maximal degree is realized by a unique node whose height is exponentially distributed and we also prove that the conditioned L{\'e}vy tree can be obtained by grafting a L{\'e}vy forest on an independent size-biased L{\'e}vy tree with a degree constraint at a uniformly chosen leaf. Finally, we show that the L{\'e}vy tree conditioned on having large maximal degree converges locally to an immortal tree (which is the continuous analogue of the Kesten tree) in the critical case and to a condensation tree in the subcritical case. Our results are formulated in terms of the exploration process which allows to drop the Grey condition.