Critical Dynamics of Random Surfaces: Time Evolution of Area and Genus

Conformal field theories with central charge $c\le1$ on random surfaces have been extensively studied in the past. Here, this discussion is extended from their equilibrium distribution to their critical dynamics. This is motivated by the conjecture that these models describe the time evolution of certain social networks that are self-driven to a critical point. This paper focuses on the dynamics of the overall area and the genus of the surface. The time evolution of the area is shown to follow a Cox Ingersol Ross process. Planar surfaces shrink, while higher genus surfaces grow to a size of order of the inverse cosmological constant. The time evolution of the genus is argued to lead to two different phases, dominated by (i) planar surfaces, and (ii) ``foamy'' surfaces, whose genus diverges. In phase (i), which exhibits critical phenomena, time variations of the order parameter are approximately t-distributed with 4 or more degrees of freedom.