Abstract
Let f be a stationary isotropic non-degenerate Gaussian field on R2. Assume that f=q∗W where q∈L2(R2)∩C2(R2) 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. 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.
Highlights
Let f be a stationary isotropic non-degenerate Gaussian field on R2
We aimed to provide a proof inspired more by the KKL inequality than randomized algorithms and presented in the native language of smooth Gaussian fields, in order to bring out the underlying mechanism
Let f be a Gaussian field on R2 of the form q ∗ W where W is the L2 white noise on R2 and q satisfies the (Strong) version of Conditions 1.1 and 1.5, as well as Conditions 1.3 1.4 and the (Weak) Condition 1.2
Summary
1.1 On previous results in Gaussian field percolation and the contributions of the present work. Otherwise, by stationarity, f would satisfy a non-trivial differential equation of the form λf + μ∂vf = 0 which would contradict the decay in pointwise correlations It was shown by Pitt in [24] that the (Weak) Condition 1.5 is equivalent to a certain form of the FKG inequality (see Lemma 4.5 below). Let f be a Gaussian field on R2 of the form q ∗ W where W is the L2 white noise on R2 and q satisfies the (Strong) version of Conditions 1.1 and 1.5, as well as Conditions 1.3 and 1.4, and the (Weak) Condition 1.2.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
