Harmonic Measure and Winding of Conformally Invariant Curves

Details Harmonic Measure and Winding of Conformally Invariant Curves Harmonic Measure and Winding of Conformally Invariant Curves

2002-08-02

arxiv.org/abs/cond-mat/0208045v1

Physics

Condensed Matter

Statistical Mechanics

3 figures

Scientific paper

10.1103/PhysRevLett.89.264101

The exact joint multifractal distribution for the scaling and winding of the electrostatic potential lines near any conformally invariant scaling curve is derived in two dimensions. Its spectrum f(alpha,lambda) gives the Hausdorff dimension of the points where the potential scales with distance $r$ as $H \sim r^{\alpha}$ while the curve logarithmically spirals with a rotation angle phi=lambda ln r. It obeys the scaling law f(\alpha,\lambda)=(1+\lambda^2) f(\bar \alpha)-b\lambda^2 with \bar \alpha=\alpha/(1+\lambda^2) and b=(25-c)/{12}$, and where f(\alpha)\equiv f(\alpha,0) is the pure harmonic measure spectrum, and c the conformal central charge. The results apply to O(N) and Potts models, as well as to {\rm SLE}_{\kappa}.

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