Log-Sobolev inequality for the sinh-Gordon model

Abstract

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. This implies that infinite-volume ``massless'' sinh-Gordon measures on ${\mathbb{R}}^2$ satisfying all Osterwalder--Schrader axioms exist and that they have exponential decay of correlations (are ``massive''). 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.

Added exponential decay and convergence of infinite-volume measure