Electronic Journal of Probability

Profile of a self-similar growth-fragmentation

François Gaston Ged

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A self-similar growth-fragmentation describes the evolution of particles that grow and split as time passes. Its genealogy yields a self-similar continuum tree endowed with an intrinsic measure. Extending results of Haas [13] for pure fragmentations, we relate the existence of an absolutely continuous profile to a simple condition in terms of the index of self-similarity and the so-called cumulant of the growth-fragmentation. When absolutely continuous, we approximate the profile by a function of the small fragments, and compute the Hausdorff dimension in the singular case. Applications to Boltzmann random planar maps are emphasized, exploiting recently established connections between growth-fragmentations and random maps [1, 5].

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 7, 21 pp.

Received: 12 April 2018
Accepted: 3 December 2018
First available in Project Euclid: 15 February 2019

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Primary: 60J25: Continuous-time Markov processes on general state spaces 60G18: Self-similar processes 60G30: Continuity and singularity of induced measures

self-similar growth-fragmentations intrinsic area profile of a tree

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Ged, François Gaston. Profile of a self-similar growth-fragmentation. Electron. J. Probab. 24 (2019), paper no. 7, 21 pp. doi:10.1214/18-EJP253. https://projecteuclid.org/euclid.ejp/1550199785

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