Quasi-independence for nodal lines

Abstract

We prove a quasi-independence result for level sets of a planar centered stationary Gaussian field with covariance (x, y) → κ(x − y), with only mild conditions on the regularity of κ. As a first application, we study percolation for nodal lines in the spirit of [BG16]. In the said article, Beffara and Gayet rely on Tassion's method ([Tas16]) to prove that, under some assumptions on κ, most notably that κ ≥ 0 and κ(x) = O(|x| −325 ), the nodal set satisfies a box-crossing property. The decay exponent was then lowered to 16 + ε by Beliaev and Muirhead in [BM18]. In the present work we lower this exponent to 4 + ε thanks to a new approach towards quasi-independence for crossing events. This approach does not rely on quantitative discretization. Our quasi-independence result also applies to events counting nodal components and we obtain a lower concentration result for the density of nodal components around the Nazarov and Sodin constant from [NS16].

The sharp phase transition for level set percolation of smooth planar Gaussian fields

“…Moreover, whereas the previous best known sufficient condition for quasi-independence was polynomial decay with exponent β > 4 [ RV19b], we deduce asymptotic independence as long as correlations decay (roughly speaking) polynomially with exponent β > 2. More precisely (in the case r = R 1 = R 2 = R), [RV19b] roughly implies that (4.2) holds for crossing events with the right-hand-side replaced by cR −β × (Total influence) 2 where the term 'Total influence' is the sum of R 2 influences defined in the spirit of the influences from Section 2.2.…”

“…Remark 1.12. The first statement of Theorem 1.11 gives analogues of the RSW estimates in critical percolation theory (see for instance [Gri99,BR06]), which have previously been established under stronger conditions on the decay of correlations (roughly corresponding to β > 4) [BG17,BM18,RV19b]. The statement about Arm 0 (r, R) is the analogue of the one-arm decay in percolation theory, and follows in a straightforward way from the RSW estimates (at least, if a preliminary 'quasi-independence' property has been established; see Theorem 4.2).…”

“…Given the discussion above, we see that our results apply to this planar Gaussian field for each β > 2. On the other hand, the covariance kernel of this field satisfies κ(x) ∼ c|x| −β , as |x| → ∞, and so none of the previous results in the literature (for instance in [BG17,RV19b]) apply to this field unless β > 4, and even then only the RSW estimates of Theorem 1.11 were known [RV19b]. In particular, the results in Theorems 1.6 and 1.11 in the case > 0, and the result in Theorem 1.15, were not previously known for any value of β > 2.…”

“…The pioneering work was [BG17] in which it was shown that, assuming correlations decay polynomially with exponent at least β ≈ 325, both L 0 and E 0 satisfy equivalents of the RSW estimates. Although the necessary decay assumptions on κ have been progressively weakened [BM18,BMW17,RV19b], the state of the art still requires correlations to decay polynomially with exponent β > 4, much faster than that implied by the mere integrability of the covariance kernel (which corresponds roughly to β > 2). For sub-critical levels < 0, [RV19a] established the exponential decay of crossing probabilities in the special case of the Bargmann-Fock field.…”

Expected number of nodal components for cut‐off fractional Gaussian fields

“…For any set , we say that an event is increasing if for any The following result is essentially due to Loren Pitt and says that Gaussian vectors with non‐negative covariance satisfy the FKG inequality. Loren Pitt stated it for finite‐dimensional Gaussian vectors but the general case follows easily (see for instance [, Theorem A.4]). Lemma Let be an a.s. continuous Gaussian random field on a separable topological space with covariance .…”