Hyperbolic sine-Gordon model beyond the first threshold

Younes Zine

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arXiv · 2504.07944

Hyperbolic sine-Gordon model beyond the first threshold

Younes Zine

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Abstract

We study the hyperbolic sine-Gordon model, with a parameter $\be^2 > 0$, and its associated Gibbs dynamics on the two-dimensional torus. By introducing a physical space approach to the Fourier restriction norm method and establishing nonlinear dispersive smoothing for the imaginary multiplicative Gaussian chaos, we construct invariant Gibbs dynamics for the hyperbolic sine-Gordon model beyond the first threshold $\be^2 = 2\pi$. The deterministic step of our argument hinges on establishing key bilinear estimates, featuring weighted bounds for cone multipliers. Moreover, the probabilistic component involves a careful analysis of the imaginary Gaussian multiplicative chaos and reduces to integrating singularities along space-time light cones. As a by-product of our proof, we identify $\be^2 = 6\pi$ as a critical threshold for the hyperbolic sine-Gordon model, which is quite surprising given that the associated parabolic model has a critical threshold at $\be^2 =8\pi$.

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Younes Zine. 2025-04-10. Hyperbolic sine-Gordon model beyond the first threshold. https://arxiv.org/abs/2504.07944

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