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Three central limit theorems for the unbounded excursion component of a Gaussian field

Michael McAuley

Annals of Applied Probability2026https://doi.org/10.1214/25-aap2228preprint
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Abstract

For a smooth, stationary Gaussian field f on Euclidean space with fast correlation decay, there is a critical level ℓc such that the excursion set {f≥ℓ} contains a (unique) unbounded component if and only if ℓ<ℓc. We prove central limit theorems for the volume, surface area and Euler characteristic of this unbounded component restricted to a growing box. For planar fields, the results hold at all supercritical levels (i.e., all ℓ<ℓc). In higher dimensions the results hold at all sufficiently low levels (all ℓ<−ℓc<ℓc) but could be extended to all supercritical levels by proving the decay of truncated connection probabilities. Our proof is based on the martingale central limit theorem.

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https://doi.org/https://doi.org/10.1214/25-aap2228

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@article{michael2026,
  title        = {{Three central limit theorems for the unbounded excursion component of a Gaussian field}},
  author       = {Michael McAuley},
  journal      = {Annals of Applied Probability},
  year         = {2026},
  doi          = {https://doi.org/https://doi.org/10.1214/25-aap2228},
}

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Three central limit theorems for the unbounded excursion component of a Gaussian field

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