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Let f be a stationary isotropic non-degenerate Gaussian field on R^2. Assumethat f = q * W where q is both C^2 and L^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, theset {f > -l} percolates with probability one when l > 0 and with probabilityzero if l < 0 or l = 0. We also prove exponential decay of crossingprobabilities and uniqueness of the unbounded cluster. To this end, we study aGaussian field g defined on the torus and establish a superconcentrationformula for the threshold T(g) which is the minimal value such that {g > -T(g)}contains a non-contractible loop. This formula follows from a GaussianTalagrand type inequality.

Talagrand's inequality in planar Gaussian field percolation