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Steep Points of Gaussian Free Fields in Any Dimension

Author

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  • Linan Chen

    (McGill University)

Abstract

This work aims to extend the existing results on the Hausdorff dimension of the classical thick point sets to a more general class of exceptional sets of a Gaussian free field (GFF). We adopt a circle or sphere averaging regularization to study a log-correlated or polynomial-correlated GFF in any dimension and introduce the notion of “f-steep points” of the GFF for a certain test function f. Roughly speaking, the f-steep points of the GFF are locations where, when weighted by the function f, the “rate of change” of the regularized field becomes unusually large. Different choices of f lead to the study of different exceptional behaviors of the GFF. We determine the Hausdorff dimension of the set consisting of f-steep points, from which not only can we recover the existing results on thick point sets for both log-correlated and polynomial-correlated GFFs, but we also obtain new results for 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.

Suggested Citation

  • Linan Chen, 2021. "Steep Points of Gaussian Free Fields in Any Dimension," Journal of Theoretical Probability, Springer, vol. 34(4), pages 1959-2004, December.
    • Handle: RePEc:spr:jotpro:v:34:y:2021:i:4:d:10.1007_s10959-020-01028-7
    DOI: 10.1007/s10959-020-01028-7
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