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Journal of Statistical Physics

, Volume 166, Issue 5, pp 1226–1246 | Cite as

The Largest Fragment of a Homogeneous Fragmentation Process

  • Andreas Kyprianou
  • Francis Lane
  • Peter MörtersEmail author
Article

Abstract

We show that in homogeneous fragmentation processes the largest fragment at time t has size
$$\begin{aligned} e^{-t \Phi '(\overline{p})}t^{-\frac{3}{2} (\log \Phi )'(\overline{p})+o(1)}, \end{aligned}$$
where \(\Phi \) is the Lévy exponent of the fragmentation process, and \(\overline{p}\) is the unique solution of the equation \((\log \Phi )'(\bar{p})=\frac{1}{1+\bar{p}}\). We argue that this result is in line with predictions arising from the classification of homogeneous fragmentation processes as logarithmically correlated random fields.

Notes

Acknowledgements

We thank Yan Fyodorov who suggested this project to us. We would like to thank three anonymous referees for their careful reading of an earlier version of this paper. Francis Lane was supported by an EPSRC studentship for the duration of this project.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of BathBathUK

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