Mathematical Background¶

Harmonic Measure¶

Definition¶

For a domain \(\Omega\) with boundary \(\partial\Omega\), the harmonic measure of a boundary subset \(E \subseteq \partial\Omega\) from a point \(z \in \Omega\) is:

\[\omega(z, E, \Omega) = u_E(z)\]

where \(u_E\) solves:

\[\begin{split}\begin{cases} \Delta u = 0 & \text{in } \Omega \\ u = \mathbf{1}_E & \text{on } \partial\Omega \end{cases}\end{split}\]

Probabilistic Interpretation¶

\(\omega(z, E, \Omega)\) equals the probability that a Brownian motion starting at \(z\) first exits \(\Omega\) through the set \(E\).

Conformal Invariance¶

Harmonic measure is conformally invariant: if \(f: \Omega \to \Omega'\) is conformal, then:

\[\omega(z, E, \Omega) = \omega(f(z), f(E), \Omega')\]

This relates harmonic measure to the derivative of the Riemann map.

Boundary Crowding¶

The Phenomenon¶

For domains approaching a polygon, boundary correspondence becomes highly non-uniform. Near corners:

Physical boundary segments of size \(O(\epsilon)\)
Map to preimage arcs of size \(O(\epsilon^{\alpha})\)
Where \(\alpha = \pi / (\pi - \theta)\) for interior angle \(\theta\)

Lamé Domains¶

For Lamé curves \(|x|^{2n} + |y|^{2n} = 1\):

As \(n \to \infty\), the curve approaches a square
Corner regions have interior angle \(\theta = \pi/2\)
So \(\alpha = \pi / (\pi - \pi/2) = 2\)

This gives the scaling law: corner arc length \(\sim n^{-2}\).

Computational Approach¶

Why Not Schwarz-Christoffel?¶

Traditional Schwarz-Christoffel methods:

Require solving for prevertices (crowded near corners)
Involve evaluating elliptic-type integrals
Can suffer from numerical instability

Direct Harmonic Measure¶

Our approach:

Solve BIE on the smooth Lamé boundary
Compute harmonic measure directly
No prevertex computation needed
Stable panel-based numerics