Log-Sobolev inequality for the sinh-Gordon model

For the sinh-Gordon model (without external mass term), we show that the log-Sobolev inequality holds uniformly in the lattice and volume regularisations for all parameters in the regime in which the Gaussian multiplicative chaos has second moments. As a consequence, this establishes that infinite-volume (massless) sinh-Gordon measures on ${\mathbb{R}}^2$ exist. The proof uses the Polchinski equation method for log-Sobolev inequalities. The required estimates on the renormalised potential are established using simple perturbation theoretic bounds, valid for absolutely monotone potentials, and an extension of the correlation inequality of Ding–Song–Sun to potentials in the GHS class. This inequality is proved also by using a variant of the Polchinski equation through the maximum principle.