A sprinkled decoupling inequality for Gaussian processes and applications

We establish the sprinkled decoupling inequality P[X is an element of A(1) boolean AND A(2)] - P [X + epsilon is an element of A(1)][X +epsilon is an element of A(2)] <= c parallel to KI1, I-2 parallel to infinity/epsilon(2) where X is an arbitrary Gaussian vector, A(1) and A(2) are increasing events that depend on coordinates I-1 and I-2 respectively, epsilon > 0 is a sprinkling parameter, parallel to KI1;I-2 parallel to infinity is the maximum absolute covariance between coordinates of X in I-1 and I-2, 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 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.