Critical Dynamics of Random Surfaces

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. The time evolution of the surface area is identified as a Cox Ingersol Ross process. Planar surfaces shrink, while higher genus surfaces grow until the cosmological constant stops their growth. Three different equilibrium states are distingushed, dominated by (i) small planar surfaces, (ii) large surfaces with high but finite genus, and (iii) foamy surfaces, whose genus diverges. Time variations of the order parameter are analyzed and are found to have generalized hyperbolic distributions. In state (i), those have power law tails with a tail index close to 4. Analogies between the time evolution of the order parameter and a multifractal random walk are also pointed out.