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On the spectral geometry of Liouville quantum gravity

Nathanaël Berestycki

TL;DR

This note surveys the spectral geometry of Liouville quantum gravity (LQG), grounding eigenvalues and eigenfunctions in the Liouville Brownian motion framework and its Green operator. It establishes a Weyl law in LQG, showing the eigenvalue counting function grows linearly with $\lambda$ up to a random scaling by the Liouville measure and a coupling-parameter dependent constant $c_\gamma$, derived via heat-trace and bridge techniques. The work also details the heat kernel decomposition, on-diagonal heat-kernel behavior, and annealed vs quenched asymptotics, and it culminates in a set of open problems linking spectral theory, boundary phenomena, and quantum chaos to random planar maps and KPZ-type scaling. Overall, the paper highlights deep connections between random geometry, spectral invariants, and potential universality classes arising in LQG and related random surfaces.

Abstract

We give a concise presentation of the construction of the Liouville quantum gravity (LQG) eigenvalues and eigenfunctions, i.e., the spectrum associated to the infinitesimal generator of Liouville Brownian motion, the canonical diffusion in the geometry of LQG. We describe the recently obtained Weyl law in this context (giving a short summary of its proof) and report on some work in progress concerning the associated heat trace. Finally, we summarise and propose some new key open problems in this direction.

On the spectral geometry of Liouville quantum gravity

TL;DR

This note surveys the spectral geometry of Liouville quantum gravity (LQG), grounding eigenvalues and eigenfunctions in the Liouville Brownian motion framework and its Green operator. It establishes a Weyl law in LQG, showing the eigenvalue counting function grows linearly with up to a random scaling by the Liouville measure and a coupling-parameter dependent constant , derived via heat-trace and bridge techniques. The work also details the heat kernel decomposition, on-diagonal heat-kernel behavior, and annealed vs quenched asymptotics, and it culminates in a set of open problems linking spectral theory, boundary phenomena, and quantum chaos to random planar maps and KPZ-type scaling. Overall, the paper highlights deep connections between random geometry, spectral invariants, and potential universality classes arising in LQG and related random surfaces.

Abstract

We give a concise presentation of the construction of the Liouville quantum gravity (LQG) eigenvalues and eigenfunctions, i.e., the spectrum associated to the infinitesimal generator of Liouville Brownian motion, the canonical diffusion in the geometry of LQG. We describe the recently obtained Weyl law in this context (giving a short summary of its proof) and report on some work in progress concerning the associated heat trace. Finally, we summarise and propose some new key open problems in this direction.

Paper Structure

This paper contains 20 sections, 19 theorems, 78 equations, 2 figures.

Table of Contents

  1. Overview and motivation
  2. Background
  3. Brief overview of Liouville quantum gravity
  4. Liouville Brownian motion
  5. Green function, spectrum
  6. Heat kernel, bridge identity and heat trace formula
  7. Heat kernel, bridge identity and heat trace formula
  8. Weyl's law in LQG
  9. Classical Weyl law
  10. Beyond Weyl's law.
  11. Statement of the result and sketch of proof
  12. Step 1.
  13. Step 2.
  14. On-diagonal heat kernel behaviour
  15. Open problems
  16. ...and 5 more sections

Key Result

Theorem 2.1

exists in probability. Furthermore the resulting measure is nonzero precisely when $\gamma \in [0,2)$, and is independent of the choice of regularisation kernel $\theta$ subject to very mild conditions.

Figures (2)

  • Figure 1: Empirical spacing distribution based on the first 2000 LQG eigenvalues (blue) versus GOE statistics (red). Picture from BW.
  • Figure 2: Heat map of $|\mathbf{f}_n|^2$ for $n = 2000$. Picture from BW.

Theorems & Definitions (34)

  • Theorem 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof : Proof of Lemma \ref{['L:occ']}
  • Definition 2.5
  • Theorem 2.1
  • Lemma 2.6
  • proof
  • Corollary 2.7
  • ...and 24 more