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Pré-Publication, Document De Travail Année : 2016

Local limits of large Galton–Watson trees rerooted at a random vertex

Benedikt Stufler

Résumé

We discuss various forms of convergence of the vicinity of a uniformly at random selected vertex in random simply generated trees, as the size tends to infinity. For the standard case of a critical Galton–Watson tree conditioned to be large the limit is the invariant random sin-tree constructed by Aldous (1991). In the condensation regime, we describe in complete generality the asymptotic local behaviour from a random vertex up to its first ancestor with large degree. Beyond this distinguished ancestor, different behaviour may occur, depending on the branching weights. In a subregime of complete condensation, we obtain convergence toward a novel limit tree, that describes the asymptotic shape of the vicinity of the full path from a random vertex to the root vertex. This includes the case where the offspring distribution follows a power law up to a factor that varies slowly at infinity.
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Dates et versions

ensl-01408155 , version 1 (03-12-2016)
ensl-01408155 , version 2 (14-02-2018)

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Benedikt Stufler. Local limits of large Galton–Watson trees rerooted at a random vertex. 2016. ⟨ensl-01408155v1⟩
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