Full extremal process in four-dimensional membrane model

Hao Ge; Xinyi Li; Jiaxi Zhao

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arXiv · 2507.19554

Full extremal process in four-dimensional membrane model

Hao Ge·Xinyi Li·Jiaxi Zhao

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Abstract

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.

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Hao Ge, Xinyi Li, Jiaxi Zhao. 2025-07-25. Full extremal process in four-dimensional membrane model. https://arxiv.org/abs/2507.19554

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