A sprinkled decoupling inequality for Gaussian processes and applications

TL;DR

This paper introduces a new decoupling inequality for Gaussian vectors that depends only on maximum pairwise correlation, enabling new results on phase transitions in Gaussian percolation models with specific correlation decay.

Contribution

It presents a sprinkled decoupling inequality for Gaussian processes and applies it to establish phase transition non-triviality for broader classes of Gaussian fields.

Findings

Decoupling inequality depends solely on maximum pairwise correlation.

Proves non-trivial phase transition for Gaussian fields with polylogarithmic decay.

Extends results to non-stationary fields and monochromatic random waves.

Abstract

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.