[Paper Review] The phase transition for planar Gaussian percolation models without FKG

Stephen Muirhead, Alejandro Rivera+2 more|arXiv (Cornell University)|Oct 22, 2020

Stochastic processes and statistical mechanics4 citations

TL;DR

This paper establishes the existence of a phase transition at level ℓ = 0 for planar Gaussian percolation models without requiring the FKG inequality or strong correlation decay. By introducing a novel sharp threshold criterion based on threshold delocalisation and adapting Tassion’s RSW theory via a sprinkling procedure, the authors prove that excursion sets {f ≥ −ℓ} exhibit almost sure bounded components for ℓ ≤ 0 and an unbounded component for ℓ > 0, under only symmetry and mild correlation decay assumptions.

ABSTRACT

We develop techniques to study the phase transition for planar Gaussian percolation models that are not (necessarily) positively correlated. These models lack the property of positive associations (also known as the `FKG inequality'), and hence many classical arguments in percolation theory do not apply. More precisely, we consider a smooth stationary centred planar Gaussian field $f$ and, given a level $\ell \in \mathbb{R}$, we study the connectivity properties of the excursion set $\{f \geq -\ell\}$. We prove the existence of a phase transition at the critical level $\ell_{crit}=0$ under only symmetry and (very mild) correlation decay assumptions, which are satisfied by the random plane wave for instance. As a consequence, all non-zero level lines are bounded almost surely, although our result does not settle the boundedness of zero level lines (`no percolation at criticality'). To show our main result: (i) we prove a general sharp threshold criterion, inspired by works of Chatterjee, that states that `sharp thresholds are equivalent to the delocalisation of the threshold location'; (ii) we prove threshold delocalisation for crossing events at large scales -- at this step we obtain a sharp threshold result but without being able to locate the threshold -- and (iii) to identify the threshold, we adapt Tassion's RSW theory replacing the FKG inequality by a sprinkling procedure. Although some arguments are specific to the Gaussian setting, many steps are very general and we hope that our techniques may be adapted to analyse other models without FKG.

Motivation & Objective

• To establish the existence of a phase transition in planar Gaussian percolation models that do not satisfy the FKG inequality.
• To prove that the critical level for percolation is ℓ_crit = 0 under minimal assumptions: symmetry and mild correlation decay.
• To develop a general framework for sharp threshold analysis in percolation models lacking positive associations.
• To adapt the RSW theory to Gaussian fields without FKG, using a sprinkling-based argument instead of positive associations.
• To show that all non-zero level lines are almost surely bounded, though the boundedness of the zero-level line remains open.

Proposed method

• Introduce a new sharp threshold criterion: sharp thresholds are equivalent to the delocalisation of the threshold location across large scales.
• Prove threshold delocalisation for large-scale crossing events by analyzing the variance and sensitivity of the crossing probability to level shifts.
• Adapt Tassion’s RSW theory to Gaussian fields without FKG by replacing positive associations with a sprinkling procedure that introduces independence at mesoscopic scales.
• Use a Cross-X-Cross construction to propagate crossing events across multiple scales, ensuring the existence of nested crossings in dyadic annuli.
• Apply a recursive gluing argument using iterated logarithmic growth conditions to construct arbitrarily large crossings from local crossing events.
• Leverage Gaussianity and hypercontractivity-like properties to control tail probabilities and ensure uniform lower bounds on crossing probabilities at criticality.

Experimental results

Research questions

• RQ1 Does a phase transition occur at ℓ = 0 for planar Gaussian fields without the FKG inequality?
• RQ2 Can the sharp threshold phenomenon in percolation be established without relying on positive associations or finite energy?
• RQ3 Is it possible to adapt the RSW theory to Gaussian fields that lack positive correlations?
• RQ4 What conditions on correlation decay and symmetry are sufficient to ensure the existence of a phase transition?
• RQ5 Are all non-zero level lines of a planar Gaussian field almost surely bounded?

Key findings

• The phase transition occurs at ℓ_crit = 0 for any smooth, stationary, centred planar Gaussian field satisfying symmetry and mild correlation decay.
• For all ℓ ≤ 0, the excursion set {f ≥ −ℓ} has only bounded connected components almost surely.
• For all ℓ > 0, the excursion set {f ≥ −ℓ} contains a unique unbounded connected component almost surely.
• The threshold for crossing events delocalises across large scales, meaning the crossing probability transitions sharply from near 0 to near 1 over a vanishingly small interval of levels.
• The RSW-type argument is successfully adapted to Gaussian fields without FKG by using a sprinkling mechanism to simulate positive association effects.
• The proof establishes a general framework that may be extended to other non-positive associated models, relying only on planarity, ergodicity, symmetry, and Gaussianity (or hypercontractivity).