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

Maximilian Nitzschner

doi:10.1214/18-ejp226

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.

Highlights

Level sets of the Gaussian free field on Zd, d ≥ 3, provide an example of a percolation model with long-range dependence
Its study goes back at least to the eighties, see [3], [12], [14], and it has attracted considerable attention recently, see for instance [9], [16], [18], [21] and [6]. It is well-known that the model undergoes a phase transition at some critical level h∗(d), which is both finite and strictly positive in all dimensions d ≥ 3
We obtain large deviation upper and lower bounds on the probability that the discrete blow-up of a compact set A ⊆ Rd gets disconnected from the boundary of the discrete blow-up of a box containing A in its interior, by the level set of a Gaussian free field below a level α, when the level set above α is in a strongly percolative regime

Summary

Introduction

Level sets of the Gaussian free field on Zd, d ≥ 3, provide an example of a percolation model with long-range dependence. Entropic repulsion results for the discrete Gaussian free field involve conditioning on the event that φ is non-negative over a subset of Zd that can be written as the discrete blow-up of a set in Rd. For instance, if DN = (N D) ∩ Zd with D a box in Rd, it is known that conditionally on {φx ≥ 0 for all x ∈ DN }, the average of the Gaussian free field over DN is repelled to a level 4g(0, 0) log N for large N , see e.g. Theorem 3.1, (3) of [10].

Notation and useful results

Disconnection lower bound

Disconnection upper bound

Entropic repulsion by disconnection