论文标题
密切相关的高斯领域的渗透I.亚临界连接概率的衰减
Percolation of strongly correlated Gaussian fields I. Decay of subcritical connection probabilities
论文作者
论文摘要
我们研究了一类强相关的高斯田地的亚临界游览集的连通性衰变。我们的主要结果表明，对于平滑的各向同性高斯田地，其协方差$ k（x）$定期与index $α\ in [0，1）$变化，$ \ {f \ le \ el \ ell \ el \ el \ el \ el \ el \ el \ el \ el \ el \ ell <ell_ el e el <ell_cc $的可能性，$ < log-asymptotic速率$c_α（\ ell_c- \ ell）^2 / k（r）$，用于显式$c_α> 0 $。如果$α= 1 $和$ \ int_0^\ infty k（x）dx = \ infty $，则log-asymptotic速率为$ c_1（\ ell_c- \ ell）^2 r（\ int_0^r k（x）dx）^{-1} $，以及$α> 1 $ $α> 1 $。 我们的发现扩展了$ \ mathbb {z}^d $，$ d \ ge 3 $的高斯自由场（GFF）的最新结果，并且可以解释为表明GFF的亚临界行为在具有协方差$ k（x）\ sim c | x | x | x |^d-2} $之间是普遍存在的。我们的结果也是证据支持物理学家的预测，即如果$α\ le 1 $，相关长度指数为$ν= 2/α$，而在$ d = 2 $中，我们严格地确定了$ν\ ge 2/α$。更普遍地，我们的方法为光滑高斯田地的各种渗透事件的大偏差分析打开了大门。 这是一系列两篇论文中的第一篇，研究了强相关的高斯田地的亚临界水平渗透，可以独立读取。
We study the decay of connectivity of the subcritical excursion sets of a class of strongly correlated Gaussian fields. Our main result shows that, for smooth isotropic Gaussian fields whose covariance kernel $K(x)$ is regularly varying at infinity with index $α\in [0, 1)$, the probability that $\{f \le \ell\}$, $\ell < \ell_c$, connects the origin to distance $R$ decays sub-exponentially in $R$ at log-asymptotic rate $c_α(\ell_c-\ell)^2 / K(R)$ for an explicit $c_α> 0$. If $α= 1$ and $\int_0^\infty K(x) dx = \infty$ then the log-asymptotic rate is $c_1 (\ell_c-\ell)^2 R (\int_0^R K(x) dx)^{-1}$, and if $α> 1$ the decay is exponential. Our findings extend recent results on the Gaussian free field (GFF) on $\mathbb{Z}^d$, $d \ge 3$, and can be interpreted as showing that the subcritical behaviour of the GFF is universal among fields with covariance $K(x) \sim c|x|^{d-2}$. Our result is also evidence in support of physicists' predictions that the correlation length exponent is $ν= 2/α$ if $α\le 1$, and in $d=2$ we establish rigorously that $ν\ge 2/α$. More generally, our approach opens the door to the large deviation analysis of a wide variety of percolation events for smooth Gaussian fields. This is the first in a series of two papers studying subcritical level-set percolation of strongly correlated Gaussian fields, which can be read independently.
