Critical Dynamics of Random Surfaces and Multifractal Scaling

The critical dynamics of conformal field theories on random surfaces is investigated beyond the previously studied dynamics of the overall area and the genus. It is found that the evolution of the order parameter in physical time performs a generalization of the multifractal random walk. Accordingly, the higher moments of time variations of the order parameter exhibit multifractal scaling. The series of Hurst exponents is computed and illustrated at the examples of the Ising-, 3-state-Potts-, and general minimal models as well as $c=1$ models on a random surface. It is noted that some of these models can replicate the observed multifractal scaling in financial markets.