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Hausdorff dimension of Liouville quantum gravity metric spaces

Determine the Hausdorff dimension of the random metric space (Σ, e^{γ X_g} g) associated with Liouville quantum gravity for general γ∈(0,2), extending beyond the special case γ=√8/3 where the dimension is known to be 4.

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Background

Gaussian multiplicative chaos yields a random area measure and an associated random metric for Liouville quantum gravity. Recent advances constructed the metric and proved tightness, but key geometric characteristics remain unknown.

The authors single out the determination of the Hausdorff dimension of the resulting metric space as the central open problem, with only the γ=√8/3 case currently settled at value 4.

References

The central open problem related to the distance is to determine the Hausdorff dimension of $(\Sigma, e{\gamma X_g} g)$, known to equal $4$ when $\gamma=\sqrt{8/3}$.

— Two Decades of Probabilistic Approach to Liouville Conformal Field Theory (2509.21053 - Rhodes et al., 25 Sep 2025) in Section 3 (Probabilistic foundations)