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Localization of Random Surfaces with Monotone Potentials and an FKG-Gaussian Correlation Inequality

Mark Sellke

Abstract

The seminal 1975 work of Brascamp-Lieb-Lebowitz initiated the rigorous study of Ginzberg-Landau random surface models. It was conjectured therein that fluctuations are localized on $\mathbb Z^d$ when $d\geq 3$ for very general potentials, matching the behavior of the Gaussian free field. We confirm this behavior for all even potentials $U:\mathbb R\to\mathbb R$ satisfying $U'(x)\geq \min(\varepsilon x,\frac{1+\varepsilon}{x})$ on $x\in \mathbb R^+$. Given correspondingly stronger growth conditions on $U$, we show power or stretched exponential tail bounds on all transient graphs, which determine the maximum field value up to constants in many cases. Further extensions include non-wired boundary conditions and iterated Laplacian analogs such as the membrane model. Our main tool is an FKG-based generalization of the Gaussian correlation inequality, which is used to dominate the finite-volume Gibbs measures by mixtures of centered Gaussian fields.

Localization of Random Surfaces with Monotone Potentials and an FKG-Gaussian Correlation Inequality

Abstract

The seminal 1975 work of Brascamp-Lieb-Lebowitz initiated the rigorous study of Ginzberg-Landau random surface models. It was conjectured therein that fluctuations are localized on when for very general potentials, matching the behavior of the Gaussian free field. We confirm this behavior for all even potentials satisfying on . Given correspondingly stronger growth conditions on , we show power or stretched exponential tail bounds on all transient graphs, which determine the maximum field value up to constants in many cases. Further extensions include non-wired boundary conditions and iterated Laplacian analogs such as the membrane model. Our main tool is an FKG-based generalization of the Gaussian correlation inequality, which is used to dominate the finite-volume Gibbs measures by mixtures of centered Gaussian fields.
Paper Structure (19 sections, 26 theorems, 49 equations)

This paper contains 19 sections, 26 theorems, 49 equations.

Table of Contents

  1. Model and Main Results
  2. Extensions
  3. Tail Bounds for Field Values on Transient Graphs.
  4. Order of the Maximum Value on $\Lambda$.
  5. Generalized Zero Boundary Conditions.
  6. Iterated Laplacian Random Surfaces.
  7. Related Work
  8. Notations and Definitions
  9. Correlation Inequalities and Confinement
  10. Preliminaries on FKG and Gaussian Correlation Inequalities
  11. The FKG-Gaussian Correlation Inequality
  12. Products of $1$-Dimensional Functions
  13. Specialization to $\nabla\phi$ Surfaces
  14. Tail Bounds for $1$-Dimensional Projections
  15. Proofs of Main Results
  16. ...and 4 more sections

Key Result

Theorem 1.1

Fix $\varepsilon>0$ and let $G$ be a percolation transient graph with $v\in V(G)$. Then as the $\varepsilon$-monotone $\vec{U}$ and vertex subset $\Lambda\subseteq V(G)$ vary, the set of $\mu_{G_{\Lambda},\vec{U}}$-laws of $\phi(v)$ is tight.

Theorems & Definitions (54)

  • Definition 1
  • Definition 2
  • Theorem 1.1
  • Theorem 1.2
  • Definition 3
  • Definition 4
  • Theorem 1.3
  • Definition 5
  • Definition 6
  • Theorem 1.4
  • ...and 44 more