Steep Points of Gaussian Free Fields in Any Dimension

This work aims to extend the existing results on the Hausdorff dimension of the classical thick point sets of a Gaussian free field (GFF) to a more general class of exceptional sets. We adopt the circle or sphere averaging regularization to treat a singular GFF in any dimension, and introduce the notion of "$f-$steep point" of the GFF for certain test function $f$. Roughly speaking, the $f-$steep points of a generic element of the GFF are locations where, when weighted by the function $f$, the "steepness", or in other words, the "rate of change" of the regularized field element becomes unusually large. Different choices of $f$ lead to the study of various exceptional behaviors of the GFF. We investigate the Hausdorff dimension of the set consisting of $f-$steep points, from which we can recover the existing results on thick point sets for both log-correlated and polynomial-correlated GFFs, and also obtain new results on exceptional sets that, to our best knowledge, have not been previously studied. Our method is inspired by the one used to study the thick point sets of the classical 2D log-correlated GFF.