Finite range decompositions of Gaussian fields with applications to level-set percolation

In a recent work [19], Muirhead has studied level-set percolation of (discrete or continuous) Gaussian fields, and has shown sharpness of the associated phase transition under the assumption that the field has a certain multiscale white noise decomposition, a variant of a finite-range decomposition. We show that a large class of Gaussian fields have such a white noise decomposition with optimal decay parameter. Examples include the discrete Gaussian free field, the discrete membrane model, and the mollified continuous Gaussian free field. This answers various questions from [19].
Our construction of the white-noise decomposition is a refinement of Bauerschmidt's construction of a finite-range decomposition [3]. In the continuous setting our construction is very similar to Bauerschmidt's, while in the discrete setting several new ideas are needed, including the use of a result by Pólya and Szegő on polynomials that take positive values on the positive real line.