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Hausdorff dimension of Liouville quantum gravity metric

Determine the Hausdorff dimension of the random metric space (Σ, e^{γ X_g} g), where X_g is the Gaussian Free Field on Σ and γ in (0,2), extending beyond the known value 4 at γ=√(8/3).

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Background

A random distance associated with the Liouville quantum gravity metric has been constructed. The intrinsic fractal dimension of this geometry is a key geometric characteristic.

While the dimension is known to be 4 in the special case γ=√(8/3), its value for general γ remains unknown and is a central question in the geometry of Liouville quantum gravity.

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)