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We derive a covariance formula for the number of excursion or level setcomponents of a smooth stationary Gaussian field on $\mathbb{R}^d$ contained incompact domains. We also present two applications of this formula: (1) forfields whose correlations are integrable we prove that the variance of thecomponent count in large domains is of volume order and give an expression forthe leading constant, and (2) for fields with slower decay of correlation wegive an upper bound on the variance which is of optimal order if correlationsare regularly varying, and improves on best-known bounds if correlations areoscillating (e.g.\ monochromatic random waves).

A covariance formula for the number of excursion set components of Gaussian fields and applications