Scaling limits for fractional polyharmonic Gaussian fields

This work is concerned with fractional Gaussian fields, i.e. Gaussian fields whose covariance operator is given by the inverse fractional Laplacian $(-\mathrm{\Delta }{)}^{-s}$ (where, in particular, we include the case $s>1$). We define a lattice discretization of these fields and show that their scaling limits – with respect to the optimal Besov space topology (up to an endpoint case) – are the original continuous fields. As a byproduct, in dimension $d<2s$, we prove the convergence in distribution of the maximum of the fields. A key tool in the proof is a sharp error estimate for the natural finite difference scheme for $(-\mathrm{\Delta }{)}^s$ under minimal regularity assumptions, which is also of independent interest.