Equality of critical parameters for percolation of Gaussian free field level-sets

Abstract

We consider upper level-sets of the Gaussian free field on ${\mathbb{Z}}^d$, for $d\ge 3$, above a given real-valued height parameter $h$. As $h$ varies, this defines a canonical percolation model with strong, algebraically decaying correlations. We prove that three natural critical parameters associated to this model, respectively describing a well-ordered subcritical phase, the emergence of an infinite cluster, and the onset of a local uniqueness regime in the supercritical phase, actually coincide. At the core of our proof lies a new interpolation scheme aimed at integrating out the long-range dependence of the Gaussian free field. Due to the strength of correlations, its successful implementation requires that we work in an effectively critical regime. Our analysis relies extensively on certain novel renormalization techniques that bring into play all relevant scales simultaneously. The approach in this article paves the way to a complete understanding of the off-critical phases for strongly correlated disordered systems.

arXiv comment66 pages, 5 figures, revised version. The current version is published at the Duke Mathematical Journal

References in corpus (26)

• Random walks on supercritical percolation clusters
• A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model
• From loop clusters and random interlacement to the free field
• Phase transition and level-set percolation for the Gaussian free field
• Soft local times and decoupling of random interlacements
• Decoupling inequalities and interlacement percolation on G x Z
• On chemical distances and shape theorems in percolation models with long-range correlations
• On decoupling inequalities and percolation of excursion sets of the Gaussian free field
• On the fragmentation of a torus by random walk
• The sharp phase transition for level set percolation of smooth planar Gaussian fields
• Random interlacements and the Gaussian free field
• Critical exponents for a percolation model on transient graphs
• Existence of phase transition for percolation using the Gaussian Free Field
• A lower bound on the two-arms exponent for critical percolation on the lattice
• The sign clusters of the massless Gaussian free field percolate on ${\mathbb{Z}}^d$, $d⩾3$ (and more)
• On the radius of Gaussian free field excursion clusters
• Upper bound on the disconnection time of discrete cylinders and random interlacements
• On macroscopic holes in some supercritical strongly dependent percolation models
• Geometry of Gaussian free field sign clusters and random interlacements
• Discretisation schemes for level sets of planar Gaussian fields
• Solidification of porous interfaces and disconnection
• Percolation in the Harmonic Crystal and Voter Model in three dimensions
• Disconnection by level sets of the discrete Gaussian free field and entropic repulsion
• Local picture and level-set percolation of the Gaussian free field on a large discrete torus
• Analyticity of Gaussian free field percolation observables
• Markovian loop clusters on graphs

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• On the radius of Gaussian free field excursion clusters
• Geometry of Gaussian free field sign clusters and random interlacements
• Excess deviations for points disconnected by random interlacements
• On bulk deviations for the local behavior of random interlacements
• Disconnection and entropic repulsion for the harmonic crystal with random conductances
• On the cost of the bubble set for random interlacements
• Analyticity of Gaussian free field percolation observables
• Percolation of strongly correlated Gaussian fields I. Decay of subcritical connection probabilities
• First passage percolation with long-range correlations and applications to random Schrödinger operators
• Random interlacement is a factor of i.i.d
• Supercritical percolation on graphs of polynomial growth
• First passage percolation, local uniqueness for interlacements and capacity of random walk
• Generating Galton-Watson trees using random walks and percolation for the Gaussian free field
• Uniqueness of the infinite connected component for the vacant set of random interlacements on amenable transient graphs
• Anatomy of a gaussian giant: supercritical level-sets of the free field on random regular graphs
• Sharpness of the Phase Transition for the Orthant Model
• Existence of an unbounded nodal hypersurface for smooth Gaussian fields in dimension $d\ge 3$
• On the chemical distance exponent for the two-sided level-set of the 2D Gaussian free field
• Percolation phase transition on planar spin systems
• On the percolative properties of the intersection of two independent interlacements