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Critical Dynamics of Random Surfaces and Multifractal Scaling

Christof Schmidhuber

TL;DR

The paper investigates the nonequilibrium dynamics of conformal field theories with $c\le 1$ on random surfaces, introducing a physical-time extension of critical dynamics that reveals multifractal scaling of the order-parameter. By decomposing the conformal factor into a constant mode and nonzero fluctuations, and working in model A dynamics, it derives a gravitationally dressed time evolution that maps onto a multifractal random walk with explicit H_n expressions. The authors provide a unified framework (analytic and stochastic) to obtain the Hurst exponents $H_n$ in terms of KPZ/Dyson–DDK data, and demonstrate the results across Ising, Potts, other unitary/non-unitary minimal models, and $c=1$ theories on random surfaces. The findings illuminate how gravity induces multifractality in dynamical scaling and suggest connections to empirical multifractal phenomena in complex systems, including potential financial-market analogies for certain non-unitary regimes.

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.

Critical Dynamics of Random Surfaces and Multifractal Scaling

TL;DR

The paper investigates the nonequilibrium dynamics of conformal field theories with on random surfaces, introducing a physical-time extension of critical dynamics that reveals multifractal scaling of the order-parameter. By decomposing the conformal factor into a constant mode and nonzero fluctuations, and working in model A dynamics, it derives a gravitationally dressed time evolution that maps onto a multifractal random walk with explicit H_n expressions. The authors provide a unified framework (analytic and stochastic) to obtain the Hurst exponents in terms of KPZ/Dyson–DDK data, and demonstrate the results across Ising, Potts, other unitary/non-unitary minimal models, and theories on random surfaces. The findings illuminate how gravity induces multifractality in dynamical scaling and suggest connections to empirical multifractal phenomena in complex systems, including potential financial-market analogies for certain non-unitary regimes.

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 models on a random surface. It is noted that some of these models can replicate the observed multifractal scaling in financial markets.

Paper Structure

This paper contains 14 sections, 84 equations, 3 figures.

Table of Contents

  1. Introduction and Summary
  2. Brief Reviews
  3. Critical Dynamics on a Static Surface
  4. Random Surfaces in Conformal Gauge
  5. Correlation Functions of the Conformal Factor
  6. Critical Dynamics of the Order Parameter
  7. Scaling Ansatz in Physical Time
  8. Dynamic Correlation Functions
  9. Multifractal Random Walk (MRW)
  10. Background vs. Physical Time
  11. Hurst Exponents on a Random Surface
  12. Algebraic Derivation
  13. Derivation in Terms of Stochastic Processes
  14. Examples: Minimal Models

Figures (3)

  • Figure 1: In the critical dynamics of random surfaces, the surface $\Sigma$ (drawn as a circle) evolves over a time interval $I$. Left: $\Sigma\times I$ in the physical metric $g=\hat{g} e^\phi$ for $\phi=0$ (gray) and for constant $\phi>0$ (blue). Right: two physical metrics of different $\phi_0(t)$ but same $\phi_0(0)$. If background time $\hat{T}$ is fixed, physical time $T_1, ..., T_4$ is a random variable, and vice versa.
  • Figure 2: Left: $\ln m_2$ as a stochastic process with decreasing variance in background time $\hat{T}$. Right: scatter plot of $\ln m_2$ vs $\ln T$. Their correlation slowly decreases from 1. A cross section at fixed physical time $\theta$ yields $\ln m_2$ as a stochastic process in physical time.
  • Figure 3: Inferring the skewed cross-sectional distribution $p_\theta(y)$ (gray area) along the vertical line of fixed physical time $\theta$ from the Gaussian distribution $p_{\hat{T}}(x)$ (blue area) of physical time for fixed background time $\hat{T}$. The dotted lines sketch those in the scatter plot in fig. 2 (right).