Abstract

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 $\alpha \in [0,1)$, the probability that $\{f\le \ell \}$, $\ell <{\ell }_c$, connects the origin to distance $R$ decays subexponentially in $R$ at log-asymptotic rate $c_{\alpha }{({\ell }_c-\ell )}^2∕K(R)$ for an explicit $c_{\alpha }>0$. If $\alpha =1$ and ${\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}_0^{\infty }K(x)dx=\infty$ then the log-asymptotic rate is $c_1{({\ell }_c-\ell )}^{2}R{\left({\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}_0^{R}K(x)dx\right)}^{-1}$, and if $\alpha >1$ 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 $\nu =2∕\alpha$ if $\alpha \le 1$, and for $d=2$ we establish rigorously that $\nu \ge 2∕\alpha$. 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 level-set percolation of strongly correlated Gaussian fields, which can be read independently.

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