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2012 Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation
Svante Janson
Probab. Surveys 9(none): 103-252 (2012). DOI: 10.1214/11-PS188

Abstract

We give a unified treatment of the limit, as the size tends to infinity, of simply generated random trees, including both the well-known result in the standard case of critical Galton–Watson trees and similar but less well-known results in the other cases (i.e., when no equivalent critical Galton–Watson tree exists). There is a well-defined limit in the form of an infinite random tree in all cases; for critical Galton–Watson trees this tree is locally finite but for the other cases the random limit has exactly one node of infinite degree.

The proofs use a well-known connection to a random allocation model that we call balls-in-boxes, and we prove corresponding theorems for this model.

This survey paper contains many known results from many different sources, together with some new results.

Citation

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Svante Janson. "Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation." Probab. Surveys 9 103 - 252, 2012. https://doi.org/10.1214/11-PS188

Information

Published: 2012
First available in Project Euclid: 8 March 2012

zbMATH: 1244.60013
MathSciNet: MR2908619
Digital Object Identifier: 10.1214/11-PS188

Subjects:
Primary: 60C50
Secondary: 05C05, 60F05, 60J80

Rights: Copyright © 2012 The Institute of Mathematical Statistics and the Bernoulli Society

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Vol.9 • 2012
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