Maximum and coupling of the sine-Gordon field

Abstract

For $0<\beta <6\pi$, we prove that the distribution of the centred maximum of the $\epsilon$-regularised continuum sine-Gordon field on the two-dimensional torus converges to a randomly shifted Gumbel distribution as $\epsilon \to 0$. 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.

arXiv commentAccepted version, to appear in Annals of Probability

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• Stochastic dynamics and the Polchinski equation: an introduction
• Multiscale coupling and the maximum of $\mathcal{P}(\varphi {)}_2$ models on the torus