Thick Points of High-Dimensional Gaussian Free Fields
Linan Chen
TL;DR
This paper extends the study of thick points in Gaussian Free Fields to higher dimensions with more singular correlations, using sphere averaging to analyze the Hausdorff dimension of these points.
Contribution
It introduces a new framework for defining and analyzing thick points in polynomial-correlated Gaussian Free Fields in dimensions higher than two.
Findings
Established Hausdorff dimension results for thick points
Extended thick point analysis to more singular, higher-dimensional fields
Used sphere averaging regularization for the analysis
Abstract
This work aims to extend the existing results on thick points of logarithmic-correlated Gaussian Free Fields to Gaussian random fields that are more singular. To be specific, we adopt a sphere averaging regularization to study polynomial-correlated Gaussian Free Fields in higher-than-two dimensions. Under this setting, we introduce the definition of thick points which, heuristically speaking, are points where the value of the Gaussian Free Field is unusually large. We then establish a result on the Hausdorff dimension of the sets containing thick points.
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Taxonomy
TopicsAnalysis of environmental and stochastic processes · Mathematical Dynamics and Fractals · Geometry and complex manifolds