A note on weak convergence results for infinite causal triangulations

Abstract

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.