Infinite Random Graphs with a view towards Quantum Gravity

In this thesis we study random planar graphs and some of the tools and techniques used to address some related combinatorial problems. We give an account of generating function methods, mainly focusing on some analytic aspects of generating functions. Namely, we discuss the so-called singularity analysis process, a technique that allows the transfer of the singular behaviour of certain functions to the asymptotic behaviour of their Taylor coefficients. Furthermore, we collect a set of theorems for the study of solutions of certain functional equations, which are frequent in combinatorial problems. As an application of random graph theory, we discuss the dynamical triangulation model and the causal dynamical triangulation model of twodimensional quantum gravity. Finally, we study the Ising model on certain infinite random trees, constructed as “thermodynamic” limits of Ising systems on finite random trees. We give a detailed description of the distribution of infinite spin configurations. As an application, we study the magnetization properties of such systems and prove that they exhibit no spontaneous magnetization. The basic reason is that the infinite tree has a certain one dimensional feature despite the fact that we prove its Hausdorff dimension to be 2. Furthermore, we obtain results on the spectral dimension of the trees. iii