Critical Dynamics of Random Surfaces and Multifractal Scaling

TL;DR

This paper explores the critical dynamics of conformal field theories on random surfaces, revealing multifractal scaling behavior in the evolution of the order parameter and computing Hurst exponents across various models, with implications for financial markets.

Contribution

It introduces a generalized multifractal random walk model for the order parameter dynamics on random surfaces, extending previous studies beyond area and genus effects.

Findings

Higher moments of order parameter variations show multifractal scaling.

Hurst exponents are computed for multiple models including Ising and Potts.

Some models replicate multifractal scaling observed in financial markets.

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.