  In a recent work (arXiv:2206.10724), 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
Muirhead's paper.
  Our construction of the white-noise decomposition is a refinement of
Bauerschmidt's construction of finite-range decomposition. 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
P\'olya and Szeg\H{o} on polynomials that take positive values on the positive
real line.
