Statistics of harmonic measure and winding of critical curves from conformal field theory

TL;DR

This paper explores the fractal properties of critical 2D curves, specifically their harmonic measure and winding angle, by linking these geometric features to conformal field theory operators and invariance principles.

Contribution

It establishes a connection between the fractal geometry of critical curves and conformal field theory, providing methods to compute harmonic measure and winding angle distributions.

Findings

Derived relations between harmonic measure, winding angle, and CFT operators.

Provided computational techniques for these geometric characteristics.

Enhanced understanding of fractal geometry in critical systems.

Abstract

Fractal geometry of random curves appearing in the scaling limit of critical two-dimensional statistical systems is characterized by their harmonic measure and winding angle. The former is the measure of the jaggedness of the curves while the latter quantifies their tendency to form logarithmic spirals. We show how these characteristics are related to local operators of conformal field theory and how they can be computed using conformal invariance of critical systems with central charge $c \leqslant 1$.