Harmonic balls in Liouville quantum gravity
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866909395034570752 |
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| author | Bou-Rabee, Ahmed Gwynne, Ewain |
| author_facet | Bou-Rabee, Ahmed Gwynne, Ewain |
| contents | Harmonic balls are domains which satisfy the mean-value property for harmonic functions. We establish the existence and uniqueness of harmonic balls on Liouville quantum gravity (LQG) surfaces using the obstacle problem formulation of Hele-Shaw flow. We show that LQG harmonic balls are neither Lipschitz domains nor LQG metric balls, and that the boundaries of their complementary connected components are Jordan curves.
We conjecture that LQG harmonic balls are the scaling limit of internal diffusion limited aggregation (IDLA) on random planar maps. In a companion paper, we prove this in the special case of mated-CRT maps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_11795 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Harmonic balls in Liouville quantum gravity Bou-Rabee, Ahmed Gwynne, Ewain Probability Mathematical Physics Analysis of PDEs Harmonic balls are domains which satisfy the mean-value property for harmonic functions. We establish the existence and uniqueness of harmonic balls on Liouville quantum gravity (LQG) surfaces using the obstacle problem formulation of Hele-Shaw flow. We show that LQG harmonic balls are neither Lipschitz domains nor LQG metric balls, and that the boundaries of their complementary connected components are Jordan curves. We conjecture that LQG harmonic balls are the scaling limit of internal diffusion limited aggregation (IDLA) on random planar maps. In a companion paper, we prove this in the special case of mated-CRT maps. |
| title | Harmonic balls in Liouville quantum gravity |
| topic | Probability Mathematical Physics Analysis of PDEs |
| url | https://arxiv.org/abs/2208.11795 |