A sprinkled decoupling inequality for Gaussian processes and applications

We establish a sprinkled decoupling inequality for increasing events of Gaussian vectors with an error that depends only on the maximum pairwise correlation. As an application we prove the non-triviality of the percolation phase transition for Gaussian fields on $\mathbb{Z}^d$ or $\mathbb{R}^d$ with (i) uniformly bounded local suprema, and (ii) correlations which decay at least polylogarithmically in the distance with exponent $\gamma>3$; this expands the scope of existing results on non-triviality of the phase transition, covering new examples such as non-stationary fields and monochromatic random waves.