Abstract
We consider a family of centered Gaussian fields on the d-dimensional unit box, whose covariance decreases logarithmically in the distance between points. We prove tightness of the recentered maximum of the Gaussian fields and provide exponentially decaying bounds on the right and left tails. We then apply this result to a version of the two-dimensional continuous Gaussian free field.
Highlights
Main result Let Y x : x ∈ [0, 1]d >0 be a family of centered Gaussian fields indexed by the ddimensional unit box [0, 1]d, where d is any positive integer
When d = 2, an example of a field satisfying (1.1) and (1.2) is the bulk of the mollified continuous Gaussian free field (MGFF), which will be defined in Section 3.1, and will be the object Set m of =
Our approach is motivated by recent advances in the study of the two dimensional discrete Gaussian free field (DGFF)
Summary
Main result Let Y x : x ∈ [0, 1]d >0 be a family of centered Gaussian fields indexed by the ddimensional unit box [0, 1]d, where d is any positive integer. Display (1.2), basic relations between the moments of Gaussian random variables and Kolmogorov’s continuity criterion (see [1, Theorem 1.4.17]) imply that the fields have continuous modifications. Since Slepian’s Lemma only allows comparison of fields with the same index set, we will add an appropriately chosen independent continuous field to the MBBM. Adding an independent continuous field to the MBBM does not change the maximum much, provided the continuous field is small and smooth enough. The phrase “absolute constant” will refer to fixed numbers that are independent of everything
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