Weak exponential metrics for high-dimensional log-correlated Gaussian fields

2025/12/06 by Hip, Andres A. Contreras, Zhuang, Zijie
Mathematics · #Geometry and complex manifolds #Random Matrices and Applications #Stochastic processes and statistical mechanics

paper · doi:10.48550/arxiv.2512.06292

Abstract

For log-correlated Gaussian fields on ℝ^d with d ≥ 2, Ding-Gwynne-Zhuang (2023) established the existence of subsequential limits of exponential metrics obtained from appropriate approximations. For γ∈ (0,√(2d)), we define a weak γ-exponential metric to be a map h ↦ D_h that assigns to a sample of a log-correlated Gaussian field h a continuous metric on ℝ^d satisfying a list of axioms. We prove that every subsequential limit of exponential metrics built from appropriate approximations of h is a weak γ-exponential metric in this sense. Moreover, we establish general properties that hold for any weak exponential metric: (1). sharp moment bounds for several natural distances; (2). optimal Hölder exponents when comparing D_h and the Euclidean metric; and (3). Hausdorff dimension and a KPZ relation. These results extend the two-dimensional Liouville quantum gravity metric theory to higher dimensions. Along the way we derive several useful properties for log-correlated Gaussian fields including the equivalence between white-noise decomposition and convolution, and a shell independence lemma.

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