The Spectral Dimension of 2D Quantum Gravity
J. Ambjorn, D. Boulatov, J. L. Nielsen, J. Rolf, Y. Watabiki
TL;DR
The paper derives a universal relation between the spectral dimension $d_s$ and the extrinsic Hausdorff dimension $D_H$ for two-dimensional quantum gravity coupled to Gaussian matter: $\frac{1}{d_s}=\frac{1}{D_H}+\frac{1}{2}$. Applying this to $D$ Gaussian fields yields $d_s=2$ for $D\le 1$ (since $D_H=\infty$) and $d_s=4/3$ for branched polymers with $D_H=4$; multicritical branched polymers give $d_s=\frac{2m}{2m-1}$, and the section also notes a caveat about the noncommutativity of limits in the definition of $RP'_V(T)$ when $R\to 0$. These results corroborate Liouville/KPZ-based expectations and numerical simulations, indicating a universal $d_s$ governed by central charge $c\le 1$. The discussion also highlights subtle choices in diffusion definitions on fluctuating geometries and potential non-commutativity between the $R\to0$ limit and the gravitational path integral.
Abstract
We show that the spectral dimension d_s of two-dimensional quantum gravity coupled to Gaussian fields is two for all values of the central charge c <= 1. The same arguments provide a simple proof of the known result d_s= 4/3 for branched polymers.