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Second Order Asymptotics for the Hard Wall Probability of the 2D Harmonic Crystal

Maximilian Fels, Oren Louidor, Tianqi Wu

TL;DR

The paper analyzes the hard-wall (positivity) probability for the two-dimensional discrete Gaussian free field on scaled domains, improving the known leading-order asymptotics by establishing a precise second-order correction. It does so by decomposing the field into an independent one-dimensional Gaussian component along the capacitor ψ_N and a conditioned remainder, then deriving tight double-exponential tail bounds for the conditioned minimum using a Ding-type argument and a detailed boundary-dilution analysis. The main result shows -log P(Ω_{V_N}^+) ≍ Cap^{W_N}(V_N) [4g (log N)^2 - θ_N log N log log N], with g = 2/π and θ_N in a fixed interval, revealing a subleading term of order (log N)(log log N) and confirming the entropic repulsion phenomenon with a quantified second-order correction. The approach leverages precise harmonic/Green-function estimates, random-walk escape probabilities, and Gaussian conditioning tools to translate geometric capacity into sharp probabilistic costs, connecting to prior work by Bolthausen–Deuschel–Giacomin and later minimum-structure analyses. Overall, the work sharpens our understanding of entropic barriers in planar DGFF and provides a framework for second-order asymptotics in related Gaussian interface models.

Abstract

We estimate the probability that the discrete Gaussian free field on a planar domain with Dirichlet boundary conditions stays positive in the bulk. Improving upon the result by Bolthausen, Deuschel and Giacomin from 2001, we derive the order of the subleading term of this probability when a sequence of discretized scale-ups of given domain and compactly included smooth bulk are considered. A main ingredient in the proof is the double exponential decay of the right tail of the centered minimum of the field in the bulk, conditioned on a certain weighted average of its values to be zero.

Second Order Asymptotics for the Hard Wall Probability of the 2D Harmonic Crystal

TL;DR

The paper analyzes the hard-wall (positivity) probability for the two-dimensional discrete Gaussian free field on scaled domains, improving the known leading-order asymptotics by establishing a precise second-order correction. It does so by decomposing the field into an independent one-dimensional Gaussian component along the capacitor ψ_N and a conditioned remainder, then deriving tight double-exponential tail bounds for the conditioned minimum using a Ding-type argument and a detailed boundary-dilution analysis. The main result shows -log P(Ω_{V_N}^+) ≍ Cap^{W_N}(V_N) [4g (log N)^2 - θ_N log N log log N], with g = 2/π and θ_N in a fixed interval, revealing a subleading term of order (log N)(log log N) and confirming the entropic repulsion phenomenon with a quantified second-order correction. The approach leverages precise harmonic/Green-function estimates, random-walk escape probabilities, and Gaussian conditioning tools to translate geometric capacity into sharp probabilistic costs, connecting to prior work by Bolthausen–Deuschel–Giacomin and later minimum-structure analyses. Overall, the work sharpens our understanding of entropic barriers in planar DGFF and provides a framework for second-order asymptotics in related Gaussian interface models.

Abstract

We estimate the probability that the discrete Gaussian free field on a planar domain with Dirichlet boundary conditions stays positive in the bulk. Improving upon the result by Bolthausen, Deuschel and Giacomin from 2001, we derive the order of the subleading term of this probability when a sequence of discretized scale-ups of given domain and compactly included smooth bulk are considered. A main ingredient in the proof is the double exponential decay of the right tail of the centered minimum of the field in the bulk, conditioned on a certain weighted average of its values to be zero.
Paper Structure (14 sections, 25 theorems, 132 equations)

This paper contains 14 sections, 25 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Setup and Results
  3. Top-level proof
  4. Discussion
  5. Preliminaries
  6. Notation
  7. Auxiliary lemmas
  8. Proof of the upper bound in Theorem \ref{['main_tail_bound']}
  9. Proof of Proposition \ref{['SS-dilution']}
  10. Proof of Lemma \ref{['variance_average_Delta']}
  11. Proof of Lemma \ref{['lemma:Ding_argument']}
  12. Proof of the lower bound in Theorem \ref{['main_tail_bound']}
  13. Proof of Section \ref{['s:proof_lower_bound']}'s lemmas
  14. Proofs of auxilary lemmas

Key Result

Lemma 1.1

For all $N$ large enough, the discrete relative capacity $\operatorname{Cap}^{W_N}(V_N) \in [c, C]$ for some constants $c, C \in (0, \infty)$ only depending on $V, W$.

Theorems & Definitions (45)

  • Lemma 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • proof : Proof of Theorem \ref{['main_result']}
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • ...and 35 more