Abstract
We prove the existence of scaling limits for the projection on the backbone of the random walks on the Incipient Infinite Cluster and the Invasion Percolation Cluster on a regular tree. We treat these projected random walks as Randomly trapped random walks (as defined in (Ann. Probab. 43 (2015) 2405–2457)) and thus describe these scaling limits as spatially subordinated Brownian motions.
Nous prouvons l’existence de la limite d’échelle pour la projection sur la lignée infinie de la marche aléatoire sur l’amas de percolation critique infini conditionné (IIC). Nous considérons aussi le cas de l’amas de percolation d’invasion d’un arbre régulier. Nous étudions ces marches projetées comme des marches aléatoires piégées de manière aléatoire (comme définies dans (Ann. Probab. 43 (2015) 2405–2457)). Nous pouvons décrire ces limites d’échelle comme des mouvements Browniens subordonnés spatialement.
Funding Statement
The first author was supported by NSF DMS1209165, BSF 2014019. The second author was supported by Fondecyt fellowship #1201090. The third author was supported by NSERC discovery grant and FRQNT jeune chercheur.
Acknowledgements
We would like to thank Louigi Addario-Berry for his very valuable input simplifying the proof of Lemma 8.1 and David Croydon for his useful answers to questions concerning fine properties of the Brownian motion on the CRT and its local time.
Citation
Gérard Ben Arous. Manuel Cabezas. Alexander Fribergh. "Backbone scaling limits for random walks on random critical trees." Ann. Inst. H. Poincaré Probab. Statist. 60 (3) 1814 - 1848, August 2024. https://doi.org/10.1214/23-AIHP1394
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