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Extreme local extrema of the sine-Gordon field

Michael Hofstetter

Abstract

We prove that for $β<6π$ the local extremal process of the massive sine-Gordon field on the unit torus in $d=2$ converges to a Poisson point process with random intensity measure ${\rm Z}^{\mathrm{SG}}(dx) \otimes e^{-αh}dh$ for some $α>0$. The proof combines existing methods for the extremal process associated to the Gaussian free field, which was introduced and studied by Biskup and Louidor, and a strong coupling between the sine-Gordon field and the Gaussian free field.

Extreme local extrema of the sine-Gordon field

Abstract

We prove that for the local extremal process of the massive sine-Gordon field on the unit torus in converges to a Poisson point process with random intensity measure for some . The proof combines existing methods for the extremal process associated to the Gaussian free field, which was introduced and studied by Biskup and Louidor, and a strong coupling between the sine-Gordon field and the Gaussian free field.

Paper Structure

This paper contains 15 sections, 34 theorems, 186 equations.

Table of Contents

  1. Introduction and main result
  2. Existing results and related work
  3. Extreme values of log-correlated fields
  4. Construction of the sine-Gordon field and coupling with the GFF
  5. Maximum of sine-Gordon field
  6. Proof of the main theorem
  7. Independent scale decomposition
  8. Convergence of the continuous field
  9. Convergence to a Poisson point process: proof of Theorem \ref{['thm:convergence-to-ppp']}
  10. Continuous field at local extrema: proof of Lemma \ref{['lem:r-local-maxima']} and Lemma \ref{['lem:r-limit-laplace-functional']}
  11. Reduction to the extremal process of $\tilde{\Phi}_s^\mathrm{SG}$: proof of Lemma \ref{['lem:reduction-to-phis']}
  12. Extremal process associated Gaussian free fields on the torus
  13. Distributional invariance of limit points and convergence to a Poisson point process
  14. Uniqueness of the intensity measure
  15. Size of the level sets

Key Result

theorem 1

There is a random measure ${\rm Z}^{\mathrm{SG}}(dx)$ on $\Omega$ with ${\rm Z}^\mathrm{SG}(\Omega) <\infty$ a.s. and ${\rm Z}^\mathrm{SG}(A)>0$ a.s. for every non-empty open set $A\subseteq \Omega$, such that for any sequence $(r_\epsilon)_\epsilon$ with $r_\epsilon \to 0$ and $r_\epsilon/\epsilon

Theorems & Definitions (54)

  • theorem 1
  • theorem 2: MR3262484
  • theorem 3: MR3262484
  • theorem 4
  • theorem 5
  • theorem 6
  • theorem 7
  • theorem 8
  • theorem 9
  • lemma 1
  • ...and 44 more