Emergence of fractional Gaussian free field correlations in subcritical long-range Ising models

We study corrections to the scaling limit of subcritical long-range Ising models with (super)-summable interactions on ${\mathbb{Z}}^d$. For a wide class of models, the scaling limit is known to be white noise, as shown by Newman (1980). In the specific case of couplings $J_{x,y}=\mathrm{\mid }x-y{\mathrm{\mid }}^{-d-\mathit{\alpha }}$, where $\mathit{\alpha }>0$ and $\mathrm{\mid }\cdot \mathrm{\mid }$ is the Euclidean norm, we find an emergence of fractional Gaussian free field correlations in appropriately renormalised and rescaled observables. The proof exploits the exact asymptotics of the two-point function, first established by Newman and Spohn (1998), together with the rotational symmetry of the interaction.