Abstract
For , 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 . 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.
Acknowledgements
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.
Citation
Roland Bauerschmidt. Michael Hofstetter. "Maximum and coupling of the sine-Gordon field." Ann. Probab. 50 (2) 455 - 508, March 2022. https://doi.org/10.1214/21-AOP1537
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