Talagrand's inequality in planar Gaussian field percolation

Y Let f be a stationary isotropic non-degenerate Gaussian field on R-2. Assume that f = q * W where q is an element of L-2 (R-2) boolean AND C-2 (R-2) and W is the L-2 white noise on R-2. We extend a result by Stephen Muirhead and Hugo Vanneuville by showing that, assuming that q * q is pointwise non-negative and has fast enough decay, the set {f >= -l} percolates with probability one when l > 0 and with probability zero if l = -T(g)} contains a non-contractible loop. This formula follows from a Gaussian Talagrand type inequality.