Let f be a stationary isotropic non-degenerate Gaussian field on . Assume that where and W is the white noise on . We extend a result by Stephen Muirhead and Hugo Vanneuville by showing that, assuming that is pointwise non-negative and has fast enough decay, the set percolates with probability one when and with probability zero if . We also prove exponential decay of crossing probabilities and uniqueness of the unbounded cluster. To this end, we study a Gaussian field g defined on the torus and establish a superconcentration formula for the threshold which is the minimal value such that contains a non-contractible loop. This formula follows from a Gaussian Talagrand type inequality.
The ideas of this paper stemmed from previous collaborations with Dmitry Beliaev, Stephen Muirhead and Hugo Vanneuville. I am grateful to the three of them for many helpful discussions. I am also thankful to Christophe Garban and Hugo Vanneuville for their comments on a preliminary version of this manuscript. Finally, I would like to thank the referee for their comments. I hope the revised version has improved in clarity.
"Talagrand’s inequality in planar Gaussian field percolation." Electron. J. Probab. 26 1 - 25, 2021. https://doi.org/10.1214/21-EJP585