Harmonic Measure of Critical Curves

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

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 a...

Harmonic measure and winding of conformally invariant curves.

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 approximately r(alpha) while the curve logarithmically spirals with a rotation angle phi=lambd...

The harmonic measure of critical Galton–Watson trees

We consider simple random walk on a critical Galton–Watson tree conditioned to have height greater than n. It is well known that the cardinality of the set of vertices of the tree at generation n is then of order n. We prove the existence of a constant β ≈ 0.78 such that the hitting distribution of the generation n in the tree by simple random walk is concentrated with high probability on a set...

Rigidity of harmonic measure

Let J be the Julia set of a conformal dynamics f . Provided that f is polynomial-like we prove that the harmonic measure on J is mutually absolutely continuous with the measure of maximal entropy if and only if f is conformally equivalent to a polynomial. This is no longer true for generalized polynomial-like maps. But for such dynamics the coincidence of classes of these two measures turns out...

Rational Harmonic Curves