Talagrand's inequality in planar Gaussian field percolation

2019/05/30 by Rivera, Alejandro · 2 citations
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1905.13317

Abstract

Let f be a stationary isotropic non-degenerate Gaussian field on R². Assume that f = q * W where q is both C² and L² and W is the L² white noise on R². 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 < 0 or l = 0. 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 T(g) which is the minimal value such that g > -T(g) contains a non-contractible loop. This formula follows from a Gaussian Talagrand type inequality.

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