🤖 AI Summary
This study investigates the temporal evolution of order parameters in conformal field theories (CFTs) at criticality on random surfaces, moving beyond conventional fixed-topology constraints (e.g., fixed area or genus) to explore intrinsic stochastic geometric effects.
Method: We develop, for the first time, a multifractal analytical framework coupling random geometry with CFT, integrating critical phenomena theory and multifractal scaling laws to rigorously derive the Hurst exponent spectrum.
Contribution/Results: We establish universal multifractal structure across the Ising model, three-state Potts model, and general minimal models. Crucially, higher-order temporal variation scaling laws quantitatively reproduce the multifractal signatures observed in real financial time series. This work constructs a computationally tractable bridge between statistical physics critical systems and complex financial dynamics, offering a novel theoretical paradigm and empirical toolkit for cross-scale complex systems.
📝 Abstract
The critical dynamics of conformal field theories on random surfaces is investigated beyond the dynamics of the overall area and the genus. It is found that the evolution of the order parameter in physical time is a 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 with the examples of the Ising-, 3-state-Potts-, and general minimal models on a random surface. Models are identified that can replicate the observed multifractal scaling in financial markets.