[Paper Review] Percolation of strongly correlated Gaussian fields I. Decay of subcritical connection probabilities

Stephen Muirhead, Franco Severo|arXiv (Cornell University)|Jun 21, 2022

Stochastic processes and statistical mechanics41 references4 citations

TL;DR

This paper establishes the sub-exponential decay rate of subcritical connection probabilities in strongly correlated Gaussian fields with regularly varying covariance kernels. Using generalized capacity and large deviation analysis, it proves that for isotropic Gaussian fields with covariance K(x) ∼ |x|⁻αL(|x|), the probability that the excursion set {f ≤ ℓ} connects the origin to distance R decays as exp(−cα(ℓc − ℓ)² / K(R)) when α ∈ [0, 1), extending known results for the Gaussian free field and supporting physics predictions on correlation length exponents.

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 $α\in [0, 1)$, 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.

Motivation & Objective

• To understand the decay rate of connectivity in subcritical excursion sets of strongly correlated Gaussian fields.
• To extend known results on the Gaussian free field (GFF) to a broader class of fields with long-range correlations.
• To rigorously establish the universality of the GFF's subcritical decay behavior for fields with covariance K(x) ∼ |x|⁻αL(|x|), α ∈ [0, 1).
• To provide a robust analytical framework based on generalized capacity and large deviation principles for smooth, isotropic Gaussian fields.
• To support and quantify physicists’ predictions on the correlation length exponent ν = 2/α for α ≤ 1, particularly in two dimensions.

Proposed method

• The authors define a generalized capacity CapK(D) for sets D ⊂ Rd using the kernel K, formulated as the inverse of the minimum energy integral over probability measures on D.
• They use the reproducing kernel Hilbert space (RKHS) associated with K to express the capacity as the minimal H-norm of functions h ≥ 1 on D.
• The analysis relies on the regular variation of the covariance kernel K(x) = |x|⁻αL(|x|) with index α ∈ [0, d), where L is slowly varying.
• A local-global decomposition of the field is introduced, replacing the harmonic-average decomposition used in GFF-specific proofs.
• The large deviation rate function for the field to have excess mean h is ½∥h∥²_H, which is linked to the capacity via the minimal energy principle.
• Asymptotic estimates for integrals involving regularly varying and slowly varying functions are derived using Karamata’s theorem and Potter’s bounds.

Experimental results

Research questions

• RQ1 How does the decay rate of subcritical connection probabilities behave for Gaussian fields with long-range correlations?
• RQ2 Is the subcritical decay behavior of the Gaussian free field universal among fields with covariance K(x) ∼ |x|⁻(d−2)?
• RQ3 What is the precise log-asymptotic rate of decay for P[Armℓ(R)] as R → ∞ in the subcritical regime?
• RQ4 Does the correlation length exponent ν satisfy ν = 2/α for α ≤ 1, and can this be rigorously established in two dimensions?
• RQ5 Can a general framework based on capacity and large deviations be developed for percolation in smooth, isotropic Gaussian fields with long-range dependence?

Key findings

• For smooth isotropic Gaussian fields with covariance K(x) ∼ |x|⁻αL(|x|), α ∈ [0, 1), the subcritical connection probability decays sub-exponentially at log-asymptotic rate cα(ℓc − ℓ)² / K(R) for an explicit cα > 0.
• When α = 1 and ∫₀^∞ K(x)dx = ∞, the decay rate is c₁(ℓc − ℓ)² R / (∫₀^R K(x)dx), showing a logarithmic correction in the denominator.
• If α > 1, the decay becomes exponential, indicating a sharp transition in behavior at α = 1.
• The result confirms that the GFF’s subcritical decay rate is universal among fields with covariance asymptotically matching the Green’s function K(x) ∼ c|x|⁻(d−2).
• In d = 2, the analysis rigorously establishes that the correlation length exponent ν satisfies ν ≥ 2/α, supporting the physics prediction ν = 2/α for α ≤ 1.
• The generalized capacity framework provides a robust, non-GFF-specific method for large deviation analysis of percolation events in smooth Gaussian fields.