Percolation of strongly correlated Gaussian fields I. Decay of subcritical connection probabilities

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 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.

46 pages. Version accepted for publication in Prob. Math. Phys

References in corpus (7)

• Random wave functions and percolation
• Equality of critical parameters for percolation of Gaussian free field level-sets
• Critical exponents for a percolation model on transient graphs
• On the radius of Gaussian free field excursion clusters
• Excess deviations for points disconnected by random interlacements
• On the cost of the bubble set for random interlacements
• Analyticity of Gaussian free field percolation observables

Cited by in corpus (2)

• First passage percolation, local uniqueness for interlacements and capacity of random walk
• Three central limit theorems for the unbounded excursion component of a Gaussian field