This work is concerned with fractional Gaussian fields, i.e. Gaussian fields whose covariance operator is given by the inverse fractional Laplacian $(-\Delta)^{-s}$ (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 $(-\Delta)^s$ under minimal regularity assumptions, which is also of independent interest.

翻译：本研究关注分数阶高斯场，即协方差算子由逆分数阶拉普拉斯算子$(-\Delta)^{-s}$给出的高斯场（特别地，我们包含$s >1$的情形）。我们定义了这些场的格点离散化，并证明其标度极限——关于最优Besov空间拓扑（直至端点情形）——是原始连续场。作为推论，在维度$d<2s$时，我们证明了场最大值依分布收敛性。证明中的关键工具是对$(-\Delta)^s$自然有限差分格式在最小正则性假设下的尖锐误差估计，该结果本身亦具有独立价值。