Berestycki and Wong (2023) proved a Weyl law for the spectrum of Liouville Brownian motion — the canonical diffusion on a Liouville quantum gravity surface: the eigenvalue counting function satisfies Nγ(λ)/λ→cγμγ(D)\mathbf{N}_\gamma(\lambda)/\lambda \to c_\gamma\mu_\gamma(D) in probability, with the explicit constant cγ=1/(π(2−γ2/2))c_\gamma = 1/(\pi(2-\gamma^2/2)) strictly larger than the Euclidean constant c0=1/(2π)c_0 = 1/(2\pi). They conjectured (Conjecture 3), as the Liouville quantum gravity analogue of Pólya's famous conjecture for Euclidean domains, that almost surely the counting function stays below its Weyl limit uniformly: Nγ(λ)≤cγμγ(D)λ\mathbf{N}_\gamma(\lambda) \le c_\gamma\mu_\gamma(D)\lambda for all λ≥0\lambda \ge 0. The conjecture remains open: it is restated as an open problem (Problem 4.7, together with Neumann and Berezin–Li–Yau-type variants) in Berestycki's December 2025 survey on the spectral geometry of Liouville quantum gravity, and the most advanced follow-up (Berestycki–Klein, June 2026, on the second term of the LQG heat trace) resolves a different conjecture of the same paper without touching the counting-function bound.

Background

Liouville quantum gravity (LQG) is a canonical model of a random two-dimensional surface, rooted in Polyakov's proposal to sum over random metrics in bosonic string theory. It is built from the Gaussian free field (GFF) — a centered Gaussian random generalized function hh on a domain D⊂R2D \subset \mathbb{R}^2 with covariance given by the Dirichlet Green's function — and from Gaussian multiplicative chaos: for a coupling constant γ∈(0,2)\gamma \in (0,2), the Liouville measure μγ\mu_\gamma is the random volume measure obtained as the limit of εγ2/2eγhε(x)dx\varepsilon^{\gamma^2/2}e^{\gamma h_\varepsilon(x)}dx as the mollification scale ε→0\varepsilon \to 0, where hεh_\varepsilon is a circle-average regularization of the field. This measure is almost surely singular with respect to Lebesgue measure and supported on a multifractal set. Liouville Brownian motion (LBM), constructed by Garban, Rhodes and Vargas, is the diffusion canonically associated with this geometry: the time change of planar Brownian motion by the positive continuous additive functional of μγ\mu_\gamma. When Brownian motion is killed upon leaving a bounded simply connected domain DD, the killed LBM admits a pure point spectrum λ0≤λ1≤…\boldsymbol{\lambda}_0 \le \boldsymbol{\lambda}_1 \le \dots (the Dirichlet eigenvalues of LQG), and the eigenvalue counting function is Nγ(λ):=∑n≥01{λn≤λ}\mathbf{N}_\gamma(\lambda) := \sum_{n \ge 0} 1_{\{\boldsymbol{\lambda}_n \le \lambda\}}.

In classical spectral geometry, Weyl's law states that for a planar Euclidean domain the counting function grows linearly, N(λ)/λ→c0∣D∣\mathbf{N}(\lambda)/\lambda \to c_0|D| with c0=1/(2π)c_0 = 1/(2\pi) (in the time parametrization used by Berestycki and Wong). Berestycki and Wong (2023) established the LQG counterpart: for γ∈(0,2)\gamma \in (0,2),

Nγ(λ)λ→λ→∞pcγ μγ(D),cγ=1π(2−γ2/2),\frac{\mathbf{N}_\gamma(\lambda)}{\lambda} \xrightarrow[\lambda \to \infty]{p} c_\gamma\,\mu_\gamma(D), \qquad c_\gamma = \frac{1}{\pi(2-\gamma^2/2)},

with convergence in probability over the law of the GFF, and the striking feature that cγ>c0c_\gamma > c_0 strictly: the random surface has spectral dimension 22 — the effective dimension seen by the diffusion, meaning the small-time return probability decays as t−ds/2t^{-d_s/2} with ds=2d_s=2 as on a smooth surface — but an anomalous proportionality constant.

A classical conjecture of Pólya asserts that for Dirichlet eigenvalues of a bounded Euclidean domain the counting function never exceeds its Weyl limit, N(λ)≤c0λ∣D∣\mathbf{N}(\lambda) \le c_0\lambda|D| for all λ\lambda. Pólya (1961) proved it for domains that tile the plane; Filonov, Levitin, Polterovich and Sher (2023) settled it for Euclidean balls; and the Li–Yau inequality (1983) establishes the analogous statement in a Cesàro-averaged (Berezin–Li–Yau) sense. The general Euclidean case remains open.

Berestycki and Wong note that in their numerical experiments the LQG counting function appears to always stay below the linear function cγμγ(D)λc_\gamma\mu_\gamma(D)\lambda predicted by the Weyl law, and put forward the random-geometry analogue of Pólya's conjecture (Conjecture 3 in their paper). They observe that the conjecture is only plausible because cγ>c0c_\gamma > c_0, i.e., the larger LQG constant gives the counting function extra room compared to the Euclidean line. In his December 2025 survey of the spectral geometry of LQG, Berestycki reposes the question as Problem 4.7, asking whether the Dirichlet and Neumann eigenvalues of Liouville Brownian motion satisfy Pólya's conjecture, or at least a version of the Berezin–Li–Yau inequalities; he notes the problem is difficult because it requires controlling all eigenvalue scales simultaneously rather than only the high-energy asymptotics, and that since cγ>c0c_\gamma > c_0 the Dirichlet case may be more tractable than the Neumann one. By contrast, the asymptotic side of the theory has advanced: Berestycki and Klein (2026) computed the second term of the expected LQG heat trace and of the almost-sure heat content, governed by a KPZ-type boundary exponent (an exponent predicted by the Knizhnik–Polyakov–Zamolodchikov scaling relations of random-surface theory, which ties boundary scaling to the Liouville coupling), and resolved a different conjecture of Berestycki–Wong concerning annealed on-diagonal heat-kernel asymptotics — but no uniform counting-function bound is known.

Problem Statement

Let D⊂R2D\subset\mathbb{R}^2 be a bounded simply connected domain, let hh be the Gaussian free field on DD with Dirichlet boundary conditions, let μγ\mu_\gamma be the associated Liouville measure (Gaussian multiplicative chaos) for γ∈(0,2)\gamma\in(0,2), and let {λn}n≥0\{\boldsymbol{\lambda}_n\}_{n\ge0} be the eigenvalues of Liouville Brownian motion killed on exiting DD, in nondecreasing order, with counting function Nγ(λ)=∑n≥01{λn≤λ}\mathbf{N}_\gamma(\lambda)=\sum_{n\ge0}1_{\{\boldsymbol{\lambda}_n\le\lambda\}} and Weyl constant cγ=1/(π(2−γ2/2))c_\gamma=1/(\pi(2-\gamma^2/2)).

Prove or disprove the LQG analogue of Pólya's conjecture (Conjecture 3 of Berestycki–Wong): with probability one,

Nγ(λ)≤cγ μγ(D) λfor all λ≥0.\mathbf{N}_\gamma(\lambda)\le c_\gamma\,\mu_\gamma(D)\,\lambda \quad\text{for all }\lambda\ge0.

A related but broader question is open Problem 4.7 of the 2025 survey, which asks whether the Dirichlet and Neumann eigenvalues of Liouville Brownian motion satisfy Pólya's conjecture, or at least a version of the Berezin–Li–Yau inequalities. That survey problem is not equivalent to the conjecture stated above: an affirmative answer via the averaged (Cesàro) branch alone, or via a Neumann spectral theory alone, would not establish the almost-sure pointwise Dirichlet bound, which remains the target problem here.

References

  1. Nathanaël Berestycki and Mo Dick Wong, "Weyl's law in Liouville quantum gravity" (2023, arXiv preprint, v3 revised 30 May 2024). DOI: 10.48550/arxiv.2307.05407; https://arxiv.org/abs/2307.05407
  2. Nathanaël Berestycki, "On the spectral geometry of Liouville quantum gravity" (2025, arXiv preprint, submitted 2 Dec 2025). DOI: 10.48550/arxiv.2512.02538; https://arxiv.org/abs/2512.02538 (restates the Pólya question as open Problem 4.7)
  3. Nathanaël Berestycki and Jakob Klein, "Spectral expansion of LQG heat trace and KPZ scaling" (2026, arXiv preprint, submitted 25 Jun 2026). DOI: 10.48550/arxiv.2606.26958; https://arxiv.org/abs/2606.26958
  4. Georg Pólya, "On the Eigenvalues of Vibrating Membranes", Proceedings of the London Mathematical Society, s3-11(1):419–433 (1961). DOI: 10.1112/plms/s3-11.1.419; https://doi.org/10.1112/plms/s3-11.1.419
  5. Nikolay Filonov, Michael Levitin, Iosif Polterovich, and David A. Sher, "Pólya's conjecture for Euclidean balls", Inventiones mathematicae 234:129–169 (2023). DOI: 10.1007/s00222-023-01198-1; https://doi.org/10.1007/s00222-023-01198-1
  6. Peter Li and Shing-Tung Yau, "On the Schrödinger equation and the eigenvalue problem", Communications in Mathematical Physics 88(3):309–318 (1983). DOI: 10.1007/bf01213210; https://doi.org/10.1007/bf01213210
  7. Christophe Garban, Rémi Rhodes, and Vincent Vargas, "Liouville Brownian motion", Annals of Probability 44(4):3076–3110 (2016). DOI: 10.1214/15-aop1042; https://doi.org/10.1214/15-aop1042
  8. Rémi Rhodes and Vincent Vargas, "Spectral dimension of Liouville quantum gravity", Annales Henri Poincaré 15:2281–2298 (2014). DOI: 10.1007/s00023-013-0308-y; https://doi.org/10.1007/s00023-013-0308-y
  9. Sebastian Andres, Naotaka Kajino, Konstantinos Kavvadias, and Jason Miller, "Two-sided heat kernel bounds for 8/3\sqrt{8/3}-Liouville Brownian motion" (2025, arXiv preprint, submitted 17 Jul 2025). DOI: 10.48550/arxiv.2507.13269; https://arxiv.org/abs/2507.13269
  10. Chunhao Cai, "Response calculus for spectral simplicity and joint eigenvalue densities" (2026, arXiv preprint; proves almost-sure spectral simplicity of Dirichlet Liouville Brownian motion). DOI: 10.48550/arxiv.2608.02459; https://arxiv.org/abs/2608.02459