Critical Dynamics of Random Surfaces and Multifractal Scaling

ArXiv ID: 2505.23928 “View on arXiv”

Authors: Christof Schmidhuber

Abstract

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.

Keywords: Conformal Field Theory, Multifractal Scaling, Hurst Exponents, Random Surfaces, Critical Dynamics, Equities (Implied)

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper is heavily mathematical, involving advanced concepts from conformal field theory, random surfaces, KPZ scaling, and deriving complex formulas for Hurst exponents; it does not contain empirical backtesting, datasets, or implementation details, instead noting a potential connection to finance in a future separate paper.

  flowchart TD
    A["Research Goal<br>Dynamics of order parameters<br>on random surfaces"] --> B["Methodology<br>Conformal Field Theory on<br>Random Surfaces"]
    B --> C["Input Data<br>Ising, 3-state Potts,<br>Minimal Models, c=1"]
    C --> D["Computation<br>Generalized Multifractal Random Walk<br>Calculation of Hurst Exponents"]
    D --> E["Outcome 1<br>Multifractal Scaling<br>in time variations"]
    D --> F["Outcome 2<br>Series of Hurst Exponents<br>computed"]
    D --> G["Outcome 3<br>Models replicate<br>financial market dynamics"]