Gaussian fluctuations of differential observables of stationary lattice fields

We study Gaussian fluctuations of differential observables of centered stationary random fields on the discrete $d$-dimensional torus of mesh $1\mathrm{/}N$. Our main result is a Central Limit Theorem for the fields $f\mapsto N^{-d\mathrm{/}2-m}{\sum }_x{\phi }_x^N(Pf)(x\mathrm{/}N)$, where $P$ is a scalar homogeneous constant-coefficient differential operator of order $m\ge 0$ with Fourier symbol $p$, e.g., $P=\mathrm{\Delta }$ and $m=2$. Under suitable assumptions on the spectral scaling limit $\gamma$ of the covariance and the normalized cumulants of order $r\ge 3$, we prove convergence in law in a negative Sobolev space to a centered Gaussian random distribution with covariance multiplier $\mathrm{\mid }p(k){\mathrm{\mid }}^2\gamma (k)$. The two factors separate the effects of the operator and the field and, notably, differentiation can compensate a low-frequency spectral singularity. For the lattice Gaussian Free Field, this gives a projected white-noise limit for the gradient. The proofs use Fourier analysis and the method of moments and cumulants, together with a tightness argument. Finally, we give two sufficient criteria for the cumulant assumption, namely a tree–graph bound and the Dobrushin uniqueness condition for finite-range Gibbs fields, and recover the white-noise limits of the centered Ising magnetization and Potts colour fluctuation fields at high temperature.