Full extremal process in four-dimensional membrane model
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915410425675776 |
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| author | Ge, Hao Li, Xinyi Zhao, Jiaxi |
| author_facet | Ge, Hao Li, Xinyi Zhao, Jiaxi |
| contents | We investigate the extremal process of four-dimensional membrane models as the size of the lattice $N$ tends to infinity. We prove the cluster-like geometry of the extreme points and the existence as well as the uniqueness of the extremal process. The extremal process is characterized by a distributional invariance property and a Poisson structure with explicit formulas. Our approach follows the same philosophy as the two-dimensional Gaussian free field (2D GFF) via the comparison with the modified branching random walk. The proofs leverage the ``Dysonization'' technique and a careful treatment of the correlation structure, which is more intricate than the 2D GFF case. As a by-product, we also obtain a simpler proof of a crucial sprinkling lemma. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_19554 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Full extremal process in four-dimensional membrane model Ge, Hao Li, Xinyi Zhao, Jiaxi Probability We investigate the extremal process of four-dimensional membrane models as the size of the lattice $N$ tends to infinity. We prove the cluster-like geometry of the extreme points and the existence as well as the uniqueness of the extremal process. The extremal process is characterized by a distributional invariance property and a Poisson structure with explicit formulas. Our approach follows the same philosophy as the two-dimensional Gaussian free field (2D GFF) via the comparison with the modified branching random walk. The proofs leverage the ``Dysonization'' technique and a careful treatment of the correlation structure, which is more intricate than the 2D GFF case. As a by-product, we also obtain a simpler proof of a crucial sprinkling lemma. |
| title | Full extremal process in four-dimensional membrane model |
| topic | Probability |
| url | https://arxiv.org/abs/2507.19554 |