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Massive SLE$_4$, massive CLE$_4$ and the massive planar GFF

Léonie Papon

Abstract

We construct a coupling between a massive GFF and a random curve in which the curve can be interpreted as the level line of the field and has the law of massive SLE$_4$. This coupling is obtained by reweighting the law of the standard coupling GFF-SLE$_4$ and our result can be seen as a conditional version of the path-integral formulation of the massive GFF. We then show that by reweighting the law of the coupling GFF-CLE$_4$ in a similar way, one obtains a coupling between a massive GFF and a random countable collection of simple loops, that we call massive CLE$_4$. Using this coupling, we relate massive CLE$_4$ to the massive Brownian loop soup with intensity $1/2$, thus proving a conjecture of Camia. As the law of the massive GFF, the laws of massive SLE$_4$ and massive CLE$_4$ are conformally covariant.

Massive SLE$_4$, massive CLE$_4$ and the massive planar GFF

Abstract

We construct a coupling between a massive GFF and a random curve in which the curve can be interpreted as the level line of the field and has the law of massive SLE. This coupling is obtained by reweighting the law of the standard coupling GFF-SLE and our result can be seen as a conditional version of the path-integral formulation of the massive GFF. We then show that by reweighting the law of the coupling GFF-CLE in a similar way, one obtains a coupling between a massive GFF and a random countable collection of simple loops, that we call massive CLE. Using this coupling, we relate massive CLE to the massive Brownian loop soup with intensity , thus proving a conjecture of Camia. As the law of the massive GFF, the laws of massive SLE and massive CLE are conformally covariant.
Paper Structure (40 sections, 28 theorems, 275 equations)

This paper contains 40 sections, 28 theorems, 275 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Outline of the proof of Theorem \ref{['theorem_mSLE_4']} and Theorem \ref{['theorem_mCLE_4']}
  4. Open problems
  5. Acknowledgements.
  6. Background
  7. Schramm-Loewner evolutions
  8. The Loewner equation and Schramm-Loewner evolutions
  9. Massive SLE$_4$
  10. Conformal loop ensembles
  11. Level lines of the Gaussian free field
  12. Definition of the GFF
  13. Coupling between an SLE$_4$ curve and a GFF
  14. Coupling between an CLE$_4$ and a GFF
  15. The Brownian loop measure and Le Jan's isomorphism theorem
  16. ...and 25 more sections

Key Result

Theorem 1.1

Let $D,a,b$ be as above, and let $\phi: D \to \mathbb{R}$ be the unique harmonic function in $D$ with boundary values $F^{(D,a,b)}$. Denote by ${\mathbb{P}}$ the law of the coupling $(h+\phi, \gamma)$ between a GFF $h+\phi$ with boundary conditions $\phi$ in $D$ and an SLE$_4$ curve $\gamma$ in $D$ where $\mathcal{Z}$ is a normalization constant. Then, under $\tilde{{\mathbb{P}}}$, Moreover, let

Theorems & Definitions (79)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Definition 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Remark 1.7
  • Theorem 1.8
  • Lemma 2.1
  • Remark 2.2
  • ...and 69 more