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The size of random fragmentation trees
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  • Published: 22 November 2007

The size of random fragmentation trees

  • Svante Janson1 &
  • Ralph Neininger2 

Probability Theory and Related Fields volume 142, pages 399–442 (2008)Cite this article

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  • 16 Citations

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Abstract

We consider the random fragmentation process introduced by Kolmogorov, where a particle having some mass is broken into pieces and the mass is distributed among the pieces at random in such a way that the proportions of the mass shared among different daughters are specified by some given probability distribution (the dislocation law); this is repeated recursively for all pieces. More precisely, we consider a version where the fragmentation stops when the mass of a fragment is below some given threshold, and we study the associated random tree. Dean and Majumdar found a phase transition for this process: the number of fragmentations is asymptotically normal for some dislocation laws but not for others, depending on the position of roots of a certain characteristic equation. This parallels the behavior of discrete analogues with various random trees that have been studied in computer science. We give rigorous proofs of this phase transition, and add further details. The proof uses the contraction method. We extend some previous results for recursive sequences of random variables to families of random variables with a continuous parameter; we believe that this extension has independent interest.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Uppsala University, PO Box 480, 751 06, Uppsala, Sweden

    Svante Janson

  2. Department of Mathematics and Computer Science, J.W. Goethe University, 60054, Frankfurt a.M, Germany

    Ralph Neininger

Authors
  1. Svante Janson
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  2. Ralph Neininger
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Corresponding author

Correspondence to Ralph Neininger.

Additional information

Research supported by an Emmy Noether fellowship of the DFG.

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Janson, S., Neininger, R. The size of random fragmentation trees. Probab. Theory Relat. Fields 142, 399–442 (2008). https://doi.org/10.1007/s00440-007-0110-1

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  • Received: 13 September 2006

  • Revised: 16 October 2007

  • Published: 22 November 2007

  • Issue Date: November 2008

  • DOI: https://doi.org/10.1007/s00440-007-0110-1

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Mathematics Subject Classification (2000)

  • Primary: 60F05
  • Secondary: 60J80
  • 60C05
  • 68P05
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