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

Michael Hofstetter

TL;DR

This work constructs a multiscale coupling between the Liouville and sinh-Gordon fields and the Gaussian free field in the $L^2$ phase on the 2D torus. Using a stochastic control framework and Polchinski renormalisation group, it expresses the non-Gaussian measures as controlled Gaussian dynamics and proves the existence of minimising drifts with uniform Sobolev bounds. The central result is a decomposition $\Phi^{\cE_\varepsilon}=\Phi^{\Delta_\varepsilon}+\Phi^{\mathrm{GFF}_\varepsilon}$ with $\Phi^{\Delta_\varepsilon}$ bounded in $H^\alpha$ for $\alpha>1$ (up to a range determined by $\beta$), uniformly in $\varepsilon$, allowing a continuum coupling and analysis of extreme values. The arguments adapt to both the Liouville and sinh-Gordon measures, with the ShG case requiring Brascamp–Lieb techniques to handle the non-monotone hyperbolic sine, and they yield continuum convergence results and corollaries on the asymptotics of the maximum field. This advances the program of Euclidean field theories in 2D by extending multiscale couplings to the sinh-Gordon model and clarifying the role of Sobolev regularity in the stochastic control approach.

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 $α>1$. 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.

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

TL;DR

This work constructs a multiscale coupling between the Liouville and sinh-Gordon fields and the Gaussian free field in the phase on the 2D torus. Using a stochastic control framework and Polchinski renormalisation group, it expresses the non-Gaussian measures as controlled Gaussian dynamics and proves the existence of minimising drifts with uniform Sobolev bounds. The central result is a decomposition with bounded in for (up to a range determined by ), uniformly in , allowing a continuum coupling and analysis of extreme values. The arguments adapt to both the Liouville and sinh-Gordon measures, with the ShG case requiring Brascamp–Lieb techniques to handle the non-monotone hyperbolic sine, and they yield continuum convergence results and corollaries on the asymptotics of the maximum field. This advances the program of Euclidean field theories in 2D by extending multiscale couplings to the sinh-Gordon model and clarifying the role of Sobolev regularity in the stochastic control approach.

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 , such that the difference is in a Sobolev space of regularity . The analysis covers the entire 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.
Paper Structure (14 sections, 20 theorems, 199 equations)

This paper contains 14 sections, 20 theorems, 199 equations.

Table of Contents

  1. Introduction
  2. Model and main results
  3. Literature
  4. Notation
  5. Stochastic control formulation
  6. Polchinski renormalisation group approach and variational formulation
  7. Existence and estimates for minimisers
  8. Existence of minimisers: proof of Proposition \ref{['prop:bd-polchinski-correspondence']}
  9. Estimates for minimising drifts: Proofs of Proposition \ref{['prop:lvshg-all-scales-drift']} and Proposition \ref{['prop:lvshg-small-scales-drift']}
  10. Liouville
  11. sinh-Gordon
  12. Proof of the main results
  13. Existence of a solutions to the Polchinski SDE for $\epsilon>0$
  14. Regularity of the multiplicative chaos

Key Result

theorem 1

Let $\beta \in (0, 4\pi)$. For $\cE \in \{\mathrm{Lv}, \mathrm{ShG}\}$ and $\epsilon>0$, there exists a process $\Phi^{\cE_\epsilon} \in C_0([0,\infty), H^{-\kappa})$ for any $\kappa >0$ such that where the difference field $\Phi^{\Delta_\epsilon}$ satisfies, for any $t_0> 0$, and moreover, for any $\alpha \in [1,2- \beta/4\pi)$ Finally, for any $t>0$, $\Phi^{\textup{GFF}_\epsilon}_0-\Phi_t^{\te

Theorems & Definitions (43)

  • theorem 1
  • corollary 1
  • remark 1
  • remark 2
  • remark 3
  • proposition 1
  • proposition 2
  • lemma 1
  • proof
  • proposition 3
  • ...and 33 more