The sharp phase transition for level set percolation of smooth planar Gaussian fields
The sharp phase transition for level set percolation of smooth planar Gaussian fields
复制标题
光滑平面高斯场水平集渗流的急剧相变
DOI：
10.1214/19-aihp1006
复制
发表时间：
2018
期刊：
影响因子：
--
通讯作者：
Hugo Vanneuville
中科院分区：
文献类型：
--
作者：
S. Muirhead;Hugo Vanneuville
我们证明了一类广泛的光滑中心平面高斯场的水平集的连通性表现出与伯努利渗流中的相变类似的零水平的相变。除了对称、正性和正则性条件外，我们还假设相关性以指数大于2的多项式衰减——大致相当于协方差核的可积性——而以前的相变仅在巴格曼-福克协方差核以超指数衰减的情况下才知道。我们还证明了相变是尖锐的，在没有进一步假设相关衰减的情况下，证明了在亚临界状态下交叉概率呈指数衰减。
We prove that the connectivity of the level sets of a wide class of smooth centred planar Gaussian fields exhibits a phase transition at the zero level that is analogous to the phase transition in Bernoulli percolation. In addition to symmetry, positivity and regularity conditions, we assume only that correlations decay polynomially with exponent larger than two -- roughly equivalent to the integrability of the covariance kernel -- whereas previously the phase transition was only known in the case of the Bargmann-Fock covariance kernel which decays super-exponentially. We also prove that the phase transition is sharp, demonstrating, without any further assumption on the decay of correlations, that in the sub-critical regime crossing probabilities decay exponentially.
Key to our methods is the white-noise representation of a Gaussian field; we use this on the one hand to prove new quasi-independence results, inspired by the notion of influence from Boolean functions, and on the other hand to establish sharp thresholds via the OSSS inequality for i.i.d. random variables, following the recent approach of Duminil-Copin, Raoufi and Tassion.
影响因子：
2.4
作者：
Beliaev D
通讯作者：
Beliaev D