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Multitype growth-fragmentation processes and planar excursions

Abstract : This work is devoted to the study of growth-fragmentation processes, in connection with planar excursions and Liouville quantum gravity. In a seminal paper, Bertoin, Budd, Curien and Kortchemski (BBCK) study the branching structure of these particle systems, as well as a particular family obtained in the scale limit in a Markov peeling process of large random maps. We first construct, on a half-planar excursion whose real part is a stable process, a signed version of the growth-fragmentation processes revealed by BBCK. We then establish the spinal decomposition of signed growth-fragmentation processes, and generalise this approach to processes with a finite number of types. We also focus on an extension to the spatial isotropic setting, where we see that a remarkable family of such processes appears in excursions away from the half-space. Finally, the last part of this thesis presents some advances towards understanding a certain space-filling SLE exploration of a quantum disc. These considerations are interpreted at the level of planar excursions through the mating-of-trees. We characterise the growth-fragmentation process for a special parameter of the Liouville measure, called pure gravity.
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Submitted on : Wednesday, July 6, 2022 - 1:05:11 PM
Last modification on : Friday, August 5, 2022 - 3:00:08 PM

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  • HAL Id : tel-03715382, version 1

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William da Silva. Multitype growth-fragmentation processes and planar excursions. Probability [math.PR]. Sorbonne Université, 2022. English. ⟨NNT : 2022SORUS035⟩. ⟨tel-03715382⟩

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