Discrete analogue of higher-dimensional Liouville quantum gravity

Determine the correct discrete counterpart in dimensions d≥3 of the continuum random geometry described by conformally flat Riemannian metrics g = e^{γΦ} g_0, where Φ is a log-correlated Gaussian field and γ∈(0,√{2d}). Identify a class of discrete structures (analogous to random planar maps in d=2) that discretizes this conformally flat GMC-based theory and supports an appropriate scaling limit.

Background

The paper situates its results within efforts to develop higher-dimensional analogues of discrete conformal geometry and Liouville quantum gravity (LQG). In d≥3, a natural continuum candidate is a conformally flat random Riemannian metric g = e^{γΦ} g_0 built from a log-correlated Gaussian field Φ and a background metric g_0.

While the authors show that random walks on certain embedded graphs (e.g., Voronoi tessellations with GMC centers) converge to Brownian motion modulo time change—offering evidence that sphere packings and Delaunay triangulations may act as discrete analogues of conformally flat metrics—the identification of a definitive discrete analog to this continuum theory, akin to random planar maps in d=2, remains unresolved.