Cluster counts of the Gaussian free field on Z^d are asymptotically Gaussian in d≥4; in d=3 the limit is Gaussian except possibly at non-degenerate critical points of the cluster density, where a non-Gaussian Hermite distribution appears.
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Limit theorems for the number of sign and level-set clusters of the Gaussian free field
Cluster counts of the Gaussian free field on Z^d are asymptotically Gaussian in d≥4; in d=3 the limit is Gaussian except possibly at non-degenerate critical points of the cluster density, where a non-Gaussian Hermite distribution appears.