Disconnection by level sets of the discrete Gaussian free field and entropic repulsion

Abstract

We derive asymptotic upper and lower bounds on the large deviation probability that the level set of the Gaussian free field on $Z^d$, d bigger or equal to three, below a given level disconnects the discrete blow-up of a compact set A from the boundary of the discrete blow-up of a box that contains A, when the level set of the Gaussian free field above this level is in a strongly percolative regime. These bounds substantially strengthen the results of arXiv:1412.3960, where A was a box and the convexity of A played an important role in the proof. We also derive an asymptotic upper bound on the probability that the average of the Gaussian free field well inside the discrete blow-up of A is above a certain level when disconnection occurs. The derivation of the upper bounds uses the solidification estimates for porous interfaces that were derived in the work arXiv:1706.07229 of A.-S. Sznitman and the author to treat a similar disconnection problem for the vacant set of random interlacements. If certain critical levels for the Gaussian free field coincide, an open question at the moment, the asymptotic upper and lower bounds that we obtain for the disconnection probability match in principal order, and conditioning on disconnection lowers the average of the Gaussian free field well inside the discrete blow-up of A, which can be understood as entropic repulsion.

arXiv comment23 pages, 1 figure, appeared in the Electronic Journal of Probability

References in corpus (6)

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• On decoupling inequalities and percolation of excursion sets of the Gaussian free field
• Disconnection, random walks, and random interlacements
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Cited by in corpus (10)

• Equality of critical parameters for percolation of Gaussian free field level-sets
• On the radius of Gaussian free field excursion clusters
• On macroscopic holes in some supercritical strongly dependent percolation models
• Excess deviations for points disconnected by random interlacements
• Disconnection and entropic repulsion for the harmonic crystal with random conductances
• Analyticity of Gaussian free field percolation observables
• Percolation of strongly correlated Gaussian fields I. Decay of subcritical connection probabilities
• Phase transition for level-set percolation of the membrane model in dimensions $d\ge 5$
• Entropic repulsion for the Gaussian free field conditioned on disconnection by level-sets
• Entropic repulsion for the occupation-time field of random interlacements conditioned on disconnection