March 2022 Maximum and coupling of the sine-Gordon field
Roland Bauerschmidt, Michael Hofstetter
Author Affiliations +
Ann. Probab. 50(2): 455-508 (March 2022). DOI: 10.1214/21-AOP1537


For 0<β<6π, we prove that the distribution of the centred maximum of the ε-regularised continuum sine-Gordon field on the two-dimensional torus converges to a randomly shifted Gumbel distribution as ε0. Our proof relies on a strong coupling at all scales of the sine-Gordon field with the Gaussian free field, of independent interest, and extensions of existing methods for the maximum of the lattice Gaussian free field.

Funding Statement

RB gratefully acknowledges funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 851682 SPINRG). MH was partially supported by the UK EPSRC Grant EP/L016516/1 for the Cambridge Centre for Analysis.


We thank Marek Biskup and Pierre-François Rodriguez for essential discussions that inspired the maximum problem studied in this article and the approach through a coupling, and for important feedback on a preliminary version of this manuscript. We thank Thierry Bodineau and Christian Webb for very helpful general discussions related to the sine-Gordon model, and we thank Ofer Zeitouni for a helpful discussion, which brought [1] to our attention.


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Roland Bauerschmidt. Michael Hofstetter. "Maximum and coupling of the sine-Gordon field." Ann. Probab. 50 (2) 455 - 508, March 2022.


Received: 1 September 2020; Revised: 1 May 2021; Published: March 2022
First available in Project Euclid: 24 March 2022

MathSciNet: MR4399156
zbMATH: 1505.60054
Digital Object Identifier: 10.1214/21-AOP1537

Primary: 60G60
Secondary: 82B41

Keywords: coupling , Gaussian free field , Maximum , sine-Gordon field

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 2 • March 2022
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