Critical Weyl law for Liouville quantum gravity

Determine whether an analogue of the subcritical Weyl law for the eigenvalue counting function holds for the critical Liouville quantum gravity Laplacian at the critical parameter gamma_c=2.

Background

For subcritical Liouville quantum gravity surfaces with parameter gammain(0,2), the paper recalls a Weyl law describing the asymptotic eigenvalue counting function and identifies the spectral dimension as two. The leading coefficient has a critical limit related to the critical Liouville quantum gravity measure.

The paper proves that the critical Liouville Brownian motion generator does not have discrete spectrum and that zero belongs to its essential spectrum. However, it does not establish an eigenvalue-counting asymptotic at criticality, leaving the corresponding critical Weyl law unresolved.

References

A curious observation is that as \gamma \to 2-, the leading order coefficient on the RHS of eq:weyl-law converges in probability to a multiple of the critical LQG measure , and it is very natural to ask whether an analogous Weyl law holds at criticality \gamma_c = 2.

— Power variations of critical Gaussian multiplicative chaos and their spectral applications  (2609.35681 - Wong, 28 Sep 2026) in Section 1, subsection “Spectral geometry of critical Liouville quantum gravity”