Extreme local extrema of the sine-Gordon field

TL;DR

This paper proves that the extremal process of the massive sine-Gordon field in two dimensions converges to a Poisson point process with a specific random intensity measure for certain parameter ranges.

Contribution

It establishes the convergence of the local extremal process of the sine-Gordon field to a Poisson point process, combining methods from Gaussian free field extremal analysis and strong coupling techniques.

Findings

Convergence of the extremal process to a Poisson point process for $eta<6\pi$

Identification of the limiting intensity measure involving ${ m Z}^{ ext{SG}}(dx)$ and exponential decay

Extension of extremal process analysis to the sine-Gordon field in 2D

Abstract

We prove that for $\beta<6\pi$ 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 ${\rm Z}^{\mathrm{SG}}(dx) \otimes e^{-\alpha h}dh$ for some $\alpha>0$. 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.