Percolation of strongly correlated Gaussian fields I. Decay of subcritical connection probabilities
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866916263010238464 |
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| author | Muirhead, Stephen Severo, Franco |
| author_facet | Muirhead, Stephen Severo, Franco |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_10723 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Percolation of strongly correlated Gaussian fields I. Decay of subcritical connection probabilities Muirhead, Stephen Severo, Franco Probability 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. |
| title | Percolation of strongly correlated Gaussian fields I. Decay of subcritical connection probabilities |
| topic | Probability |
| url | https://arxiv.org/abs/2206.10723 |