Growth of Lévy trees

By Thomas Duquesne, Matthias Winkel. 2005

83 citations recorded by the source corpus.

We construct random locally compact real trees called Lévy trees that are the genealogical trees associated with continuous-state branching processes. More precisely, we define a growing family of discrete Galton–Watson trees with i.i.d. exponential branch lengths that is consistent under Bernoulli percolation on leaves; we define the Lévy tree as the limit of this growing family with respect to the Gromov–Hausdorff topology on metric spaces. This elementary approach notably includes supercritical trees and does not make use of the height process introduced by Le Gall and Le Jan to code the genealogy of (sub)critical continuous-state branching processes. We construct the mass measure of Lévy trees and we give a decomposition along the ancestral subtree of a Poisson sampling directed by the mass measure.

Subfields: Condensed Matter Physics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a25d1c53c6b44332200e91688f0d490b0128a921

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In Condensed Matter Physics

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Citations in the corpus

Regenerative tree growth: Binary self-similar continuum random trees and Poisson--Dirichlet compositions (2008, unknown): “from [7] that we can associate a real tree as a subset of Tn× [0,∞) as: T = {([n], 0)} ∪ {(B, s) : B ∈ Tn, s ∈ (lB , rB ]} , (22)”
A continuum-tree-valued Markov process (2009, unknown): “Second case.”
Population Dynamics and Random Genealogies (2008, background): “Population Dynamics and Random Genealogies 151 called Lévy trees[31,33].”