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Garban–Vargas Gaussian-multiplicative-chaos Fourier-decay conjecture

Let μγ,GMC\mu_{\gamma,\mathrm{GMC}} be the one-dimensional Gaussian multiplicative chaos measure on the circle associated with a log-correlated Gaussian field. The Garban--Vargas conjecture asserts that, for 0<γ<1/20<\gamma<1/\sqrt{2}, almost surely dim⁡F(μγ,GMC)=1−γ2\dim_F(\mu_{\gamma,\mathrm{GMC}})=1-\gamma^2, where dim⁡F(μ)=sup⁡{s≥0: ∣μ^(n)∣=O(∣n∣−s/2) as ∣n∣→∞}\dim_F(\mu)=\sup\{s\geq 0:\ |\widehat{\mu}(n)|=O(|n|^{-s/2})\text{ as }|n|\to\infty\} for Fourier coefficients on the circle.

References

Primary source

arXiv

Additional references

fractal-geometryharmonic-analysisprobability
Posted · edited

Progress summary

Refreshed
Claimed solved

An unrefereed 2026 manuscript claims to settle the circle case of the Garban–Vargas conjecture, while broader versions remain outside the reported result.

Garban and Vargas conjectured a sharp formula for the Fourier dimension of one-dimensional Gaussian multiplicative chaos, building on their earlier proof that the measure is Rajchman. The conjectured value is 1−γ21-\gamma^2 for small parameters.

Known results

  • Garban and Vargas proved the Rajchman property for 0<γ<20<\gamma<\sqrt{2}.
  • For 0<γ<1/20<\gamma<1/\sqrt{2}, they established 1/2−γ2≤dim⁡F(Mγ)≤1−γ21/2-\gamma^2\leq\dim_F(M_\gamma)\leq1-\gamma^2.
  • Lin, Qiu, and Tan’s 2024 preprint claims the exact formula dim⁡F(μγ,GMC)=Dγ\dim_F(\mu_{\gamma,\mathrm{GMC}})=D_\gamma throughout the subcritical range.

September 2026 circle resolution

On September 10, 2026, a report identified Orsoni and Verreault’s manuscript as resolving the circle instance within a broader treatment of Fourier decay for chaos measures and boundary sets. The result remains unrefereed; independent coverage likewise describes the earlier claimed proof as not peer-reviewed.

Current status (as of September 2026): The circle or one-dimensional case is claimed solved, but the claim is unverified and the broader conjectural scope remains open.

Sources

Solutions 0

No solutions have been posted yet.