Steep Points of Gaussian Free Fields in Any Dimension

Linan Chen ()
Additional contact information
Linan Chen: McGill University

Journal of Theoretical Probability, 2021, vol. 34, issue 4, 1959-2004

Abstract: 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.

Keywords: Gaussian free field; Regularization processes; Steep point; Thick point; Exceptional set; Hausdorff dimension; 60G60; 60G15 (search for similar items in EconPapers)
Date: 2021
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10959-020-01028-7 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jotpro:v:34:y:2021:i:4:d:10.1007_s10959-020-01028-7

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10959

DOI: 10.1007/s10959-020-01028-7

Access Statistics for this article

Journal of Theoretical Probability is currently edited by Andrea Monica

More articles in Journal of Theoretical Probability from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().