A note on weak convergence results for infinite causal triangulations

We discuss infinite causal triangulations and equivalence to the size biased branching process measure-the critical Galton-Watson branching process distribution conditioned on non-extinction. Using known results from the theory of branching processes, this relation is used to prove a novel weak convergence result of the joint length-area process of a infinite causal triangulations to a limiting diffusion. The diffusion equation enables us to determine the physical Hamiltonian and Green's function from the Feynman-Kac procedure, providing us with a mathematical rigorous proof of certain scaling limits of causal dynamical triangulations.