  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
sub-exponentially in $R$ at log-asymptotic rate $c_\alpha (\ell_c-\ell)^2 /
K(R)$ for an explicit $c_\alpha > 0$. If $\alpha = 1$ 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 $\alpha > 1$ 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 in $d=2$ 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 subcritical level-set
percolation of strongly correlated Gaussian fields, which can be read
independently.
