Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Tightness of the maximum of Ginzburg-Landau fields

Florian Schweiger, Wei Wu, Ofer Zeitouni

Abstract

We consider the discrete Ginzburg-Landau field with potential satisfying a uniform convexity condition, in the critical dimension $d=2$, and prove that its maximum over boxes of sidelength $N$, centered by an explicit $N$-dependent centering, is tight.

Tightness of the maximum of Ginzburg-Landau fields

Abstract

We consider the discrete Ginzburg-Landau field with potential satisfying a uniform convexity condition, in the critical dimension , and prove that its maximum over boxes of sidelength , centered by an explicit -dependent centering, is tight.
Paper Structure (29 sections, 41 theorems, 350 equations)

This paper contains 29 sections, 41 theorems, 350 equations.

Table of Contents

  1. Introduction and main results
  2. Main result
  3. Earlier work
  4. Outline of the proof
  5. A conjecture
  6. Set-up and important tools
  7. Notation
  8. Brascamp-Lieb inequalities and rough upper bounds
  9. Miller's coupling
  10. The Helffer-Sjöstrand representation
  11. Quantitative CLT
  12. Random walk approximation
  13. Square averages
  14. A single coupling step
  15. Iterated coupling
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let the sequence of measures $\mathbb{P}^{Q_N,0}$ be as defined in e:defGLmeasure, and suppose that $V"$ is Lipschitz-continuous. Let $\phi_N$ be a sample from $\mathbb{P}^{Q_N,0}$. Then there is a deterministic constant $\mathbf{g}=\mathbf{g}(V)$, such that with $m_N$ as in e:defmN, for every $\var

Theorems & Definitions (79)

  • Theorem 1.1
  • Conjecture 1.2
  • Lemma 2.1: Brascamp-Lieb inequalities
  • Lemma 2.2
  • proof
  • Theorem 2.3: M11
  • Theorem 2.4: M11
  • Theorem 2.5
  • proof
  • Lemma 2.6
  • ...and 69 more