We develop a transfer matrix formalism for two-dimensional pure gravity. By taking the continuum limit, we obtain a "Hamiltonian formalism" in which the geodesic distance plays the role of time. Applying this formalism, we obtain a universal function which describes the fractal structures of two dimensional quantum gravity in the continuum limit.

The 2-point function is the natural object in quantum gravity for extracting critical behavior: The exponential fall off of the 2-point function with geodesic distance determines the fractal dimension d H of space-time. The integral of the 2-point function determines the entropy exponent γ, i.e. the fractal structure related to baby universes, while the short distance behavior of the 2-point function connects γ and d H by a quantum gravity version of Fisher's scaling relation. We verify this behavior in the case of 2d gravity by explicit calculation.

We provide evidence that the Hausdorff dimension is 4 and the spectral dimension is 2 for two-dimensional quantum gravity coupled the matter with a central charge $c \leq 1$. For $c > 1$ the Hausdorff dimension and the spectral dimension monotonously decreases to 2 and 1, respectively.Comment: 30 pages, postscript, including 11 figures, csh file.name should uudecode et

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.

Causal Dynamical Triangulations (CDT) is a lattice theory where aspects of quantum gravity can be studied. Two-dimensional CDT can be solved analytically and the continuum (quantum) Hamiltonian obtained. In this article we show that this continuum Hamiltonian is the one obtained by quantizing twodimensional projectable Hořava-Lifshitz gravity. 1 arXiv:1302.6359v1 [hep-th]