The Liouville model in the $L^1$ phase: coupling and extreme values

TL;DR

This paper establishes a strong coupling between the Liouville model and the Gaussian free field on a 2D torus in the $L^1$ phase, enabling analysis of extreme values and showing the maximum converges to a shifted Gumbel distribution.

Contribution

It introduces a novel coupling method for the Liouville model and Gaussian free field using Polchinski renormalisation, revealing new insights into extreme value behavior.

Findings

Coupling difference is Hölder continuous.

Maximum of Liouville field converges to a shifted Gumbel distribution.

Polchinski flow has a definite sign and is well-controlled.

Abstract

We establish a strong coupling between the Liouville model and the Gaussian free field on the two dimensional torus in the $L^1$ phase $\beta \in (0, 8\pi)$, such that the difference of the two fields is a H\"older continuous function. The coupling originates from a Polchinski renormalisation group approach, which was previously used to prove analogous results for other Euclidean field theories in dimension two. Our main observations for the Liouville model are that the Polchinski flow has a definite sign and can be controlled well thanks to an FKG argument. The coupling allows to relate extreme values of the Liouville model and the Gaussian free field, and as an application we show that the global maximum of the Liouville field converges in distribution to a randomly shifted Gumbel distribtion.