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Exact large-scale fluctuations of the phase field in the sine-Gordon model

We present the first exact theory and analytical formulas for the large-scale phase fluctuations in the sine-Gordon model, valid in all regimes of the field theory, for arbitrary temperatures and interaction strengths. Our result is based on the Ballistic Fluctuation Theory combined with Generalized Hydrodynamics, and can be seen as an exact ``dressing" of the phenomenological soliton-gas picture first introduced by Sachdev and Young [S. Sachdev and A. P. Young, PRL 78, 2220 (1997)], to the modes of Generalised Hydrodynamics. The resulting physics of phase fluctuations in the sine-Gordon model is qualitatively different, as the stable quasi-particles of integrability give coherent ballistic propagation instead of diffusive spreading. We provide extensive numerical checks of our analytical predictions within the classical regime of the field theory by using Monte Carlo methods. We discuss how our results are of ready applicability to experiments on tunnel-coupled quasicondensates.