  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.
