Quenched scaling limit of critical percolation clusters on Galton-Watson trees
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910114015870976 |
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| author | Archer, Eleanor Lions, Tanguy |
| author_facet | Archer, Eleanor Lions, Tanguy |
| contents | We consider quenched critical percolation on a supercritical Galton--Watson tree with either finite variance or $α$-stable offspring tails for some $α\in (1,2)$. We show that the GHP scaling limit of a quenched critical percolation cluster on this tree is the corresponding $α$-stable tree, as is the case in the annealed setting. As a corollary we obtain that a simple random walk on the cluster also rescales to Brownian motion on the stable tree. Along the way, we also obtain quenched asymptotics for the tail of the cluster size, which completes earlier results obtained in Michelen (2019) and Archer-Vogel (2024). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_12088 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quenched scaling limit of critical percolation clusters on Galton-Watson trees Archer, Eleanor Lions, Tanguy Probability 60F17, 60D05 We consider quenched critical percolation on a supercritical Galton--Watson tree with either finite variance or $α$-stable offspring tails for some $α\in (1,2)$. We show that the GHP scaling limit of a quenched critical percolation cluster on this tree is the corresponding $α$-stable tree, as is the case in the annealed setting. As a corollary we obtain that a simple random walk on the cluster also rescales to Brownian motion on the stable tree. Along the way, we also obtain quenched asymptotics for the tail of the cluster size, which completes earlier results obtained in Michelen (2019) and Archer-Vogel (2024). |
| title | Quenched scaling limit of critical percolation clusters on Galton-Watson trees |
| topic | Probability 60F17, 60D05 |
| url | https://arxiv.org/abs/2501.12088 |