On the radius of Gaussian free field excursion clusters

Abstract

We consider the Gaussian free field $\phi$ on ${\mathbb{Z}}^d$, for $d\ge 3$, and give sharp bounds on the probability that the radius of a finite cluster in the excursion set $\{\phi \ge h\}$ exceeds a large value $N$, for any height $h\ne h_{\ast }$, where $h_{\ast }$ refers to the corresponding percolation critical parameter. In dimension $d=3$, we prove that this probability is sub-exponential in $N$ and decays as $\exp \{-\frac{\pi }{6}(h-h_{\ast }{)}^2\frac{N}{\log N}\}$ as $N\to \infty$ to principal exponential order. When $d\ge 4$, we prove that these tails decay exponentially in $N$. Our results extend to other quantities of interest, such as truncated two-point functions and the two-arms probability for annuli crossings at scale N.

50 pages, revised version

References in corpus (5)

• Critical exponents for a percolation model on transient graphs
• Cluster capacity functionals and isomorphism theorems for Gaussian free fields
• Disconnection and entropic repulsion for the harmonic crystal with random conductances
• Analyticity of Gaussian free field percolation observables
• The critical two-point function for long-range percolation on the hierarchical lattice

Cited by in corpus (9)

• Critical exponents for a percolation model on transient graphs
• Disconnection and entropic repulsion for the harmonic crystal with random conductances
• Analyticity of Gaussian free field percolation observables
• On the cost of the bubble set for random interlacements
• Percolation of strongly correlated Gaussian fields I. Decay of subcritical connection probabilities
• Phase transition for level-set percolation of the membrane model in dimensions $d\ge 5$
• First passage percolation with long-range correlations and applications to random Schrödinger operators
• 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