Hausdorff dimension of standard Liouville quantum gravity surfaces

Determine the Hausdorff dimension d(γ) of surfaces described by standard γ-Liouville quantum gravity for γ<2, thereby resolving the stated open question and assessing the conjectured Ding–Gwynne dimension formula.

Background

The paper relates the Hausdorff dimension \widetilde{d}(\gamma') of a single large bubble in the critical regime \gamma'>2 to the Hausdorff dimension d(\gamma) of standard \gamma-Liouville quantum gravity through \widetilde{d}(\gamma')/\gamma'=d(\gamma)/\gamma, with \gamma\gamma'=4. Although the authors derive the corresponding dual relation, they state that the underlying dimension d(\gamma) for standard \gamma-LQG remains unknown.

The paper also records a formula conjectured by Ding and Gwynne, which is invariant under Liouville duality. Establishing that formula for d(\gamma) would simultaneously yield the corresponding extension to the dual regime \gamma'>2.