Exact multifractal spectra for arbitrary laplacian random walks.

Area of Science:

• * Statistical physics and complex systems analysis.
• * Two-dimensional conformal field theory and fractal geometry.

Background:

• * Understanding the multifractal properties of random walks is crucial in various scientific fields.
• * Laplacian random walks exhibit complex scaling behaviors that require advanced analytical techniques.

Purpose of the Study:

• * To derive exact multifractal spectra for harmonic measures of arbitrary Laplacian random walks in two dimensions.
• * To analyze the distinct scaling behaviors of the growth measure in different spatial and temporal regimes.

Main Methods:

• * Utilizing iterated conformal mappings as the primary analytical tool.
• * Decomposing the multifractal spectra into components related to time scaling, proximity to the growth tip, and regions distant from the tip.

Main Results:

• * Identified separate multifractal spectra for the growth measure in time, near the growth tip, and away from the growth tip.
• * Demonstrated that spectra away from the tip align with conformally invariant equilibrium systems (central charge c ≤ 1).
• * Showed that time-dependent scaling and scaling near the tip deviate from equilibrium properties.

Conclusions:

• * Iterated conformal mappings provide a powerful method for characterizing multifractality in Laplacian random walks.
• * The study reveals a dichotomy in scaling behavior: equilibrium-like properties away from the tip and non-equilibrium dynamics near the tip and in time.
• * Findings offer new insights into the statistical properties of random processes in two dimensions.