A sprinkled decoupling inequality for Gaussian processes and applications

We establish the sprinkled decoupling inequality (Formula Presented) where X is an arbitrary Gaussian vector, A₁ and A₂ are increasing events that depend on coordinates I₁ and I₂ respectively, ϵ > 0 is a sprinkling parameter, kK_I1;I₂ ꝏ is the maximum absolute covariance between coordinates of X in I₁ and I₂, and c > 0 is a universal constant. As an application we prove the non-triviality of the percolation phase transition for Gaussian fields on Z^d or R^d with (i) uniformly bounded local suprema, and (ii) correlations which decay at least polylogarithmically in the distance with exponent γ > 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.