We discuss a family of random fields indexed by a parameter s ∈ R which we call the fractional Gaussian fields, given bywhere W is a white noise on R d and (−∆) −s/2 is the fractional Laplacian. These fields can also be parameterized by their Hurst parameter H = s − d/2. In one dimension, examples of FGF s processes include Brownian motion (s = 1) and fractional Brownian motion (1/2 < s < 3/2). Examples in arbitrary dimension include white noise (s = 0), the Gaussian free field (s = 1), the bi-Laplacian Gaussian field (s = 2), the log-correlated Gaussian field (s = d/2), Lévy's Brownian motion (s = d/2 + 1/2), and multidimensional fractional Brownian motion (d/2 < s < d/2 + 1). These fields have applications to statistical physics, early-universe cosmology, finance, quantum field theory, image processing, and other disciplines.We present an overview of fractional Gaussian fields including covariance formulas, Gibbs properties, spherical coordinate decompositions, restrictions to linear subspaces, local set theorems, and other basic results. We also define a discrete fractional Gaussian field and explain how the FGF s with s ∈ (0, 1) can be understood as a long range Gaussian free field in which the potential theory of Brownian motion is replaced by that of an isotropic 2s-stable Lévy process.

We prove that the linear statistics of eigenvalues of β-log gasses satisfying the onecut and off-critical assumption with a potential V ∈ C 6 (R) satisfy a central limit theorem at all mesoscopic scales α ∈ (0; 1). We prove this for compactly supported test functions f ∈ C 5 (R) using loop equations at all orders along with rigidity estimates.

We study linear spectral statistics of N × N Wigner random matrices H on mesoscopic scales. Under mild assumptions on the matrix entries of H, we prove that after centering and normalizing, the trace of the resolvent Tr(H − z) −1 converges to a stationary Gaussian process as N → ∞ on scales N −1/3 ≪ Im(z) ≪ 1 and explicitly compute the covariance structure. The limit process is related to certain regularizations of fractional Brownian motion and logarithmically correlated fields appearing in [34]. Finally, we extend our results to general mesoscopic linear statistics and prove that the limiting covariance is given by the H 1/2 -norm of the test functions.

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We prove that Kendall's Rank correlation matrix converges to the Marčenko Pastur law, under the assumption that observations are i.i.d random vectors X1, . . . , Xn with components that are independent and absolutely continuous with respect to the Lebesgue measure. This is the first result on the empirical spectral distribution of a multivariate U -statistic.