A coupling for the Liouville and the sinh-Gordon model in the $L^2$ phase

TL;DR

This paper develops a stochastic control method to couple the Liouville and sinh-Gordon fields with the Gaussian free field in two dimensions, ensuring their difference has Sobolev regularity in the entire $L^2$ phase.

Contribution

It introduces a novel coupling technique for these fields in the $L^2$ phase using variational estimates and inequalities, extending previous results.

Findings

Couplings are established with difference in Sobolev space of regularity $eta>1$.

The approach applies to the entire $L^2$ phase, covering all relevant parameter regimes.

Utilizes estimates of minimizers and Brascamp-Lieb inequality for analysis.

Abstract

Using a stochastic control approach we establish couplings of the Liouville field and the sinh-Gordon field with the Gaussian free field in dimension $d=2$, such that the difference is in a Sobolev space of regularity $\alpha>1$. The analysis covers the entire $L^2$ phase. Our main tools are estimates for the short scales of the minimiser of the variational problem and several applications of the Brascamp-Lieb inequality.