harmonic-measure

Harmonic measure computation and boundary crowding diagnostics for 2D domains.

Overview

Harmonic measure quantifies the "visibility" of boundary subsets from interior points in potential-theoretic terms. This package provides:

• Direct computation of harmonic measure via boundary integral methods
• Corner detection for domains approaching polygonal limits
• Scaling diagnostics to quantify boundary crowding
• Visualization of harmonic measure density on boundaries

Installation

pip install harmonic-measure

# With plotting
pip install harmonic-measure[plot]

Quick Start

from harmonic_measure import HarmonicMeasure
from superellipse import Superellipse

# Create a Lamé domain approaching a rectangle
curve = Superellipse(a=1, b=1, p=32)  # p=2n, so n=16

# Compute harmonic measure of a boundary subset
hm = HarmonicMeasure(curve)
omega = hm.measure_subset(
    basepoint=[0, 0],
    subset_fn=lambda pt: abs(pt[0] - 1) < 0.1  # near corner
)
print(f"Harmonic measure of corner neighborhood: {omega:.6f}")

Theory

Harmonic Measure

For a domain Ω with boundary ∂Ω, the harmonic measure ω_Ω(z, A) of a boundary subset A ⊂ ∂Ω as seen from z ∈ Ω is:

$$\omega_\Omega(z, A) = u_A(z)$$

where u_A solves: Δu = 0 in Ω, u = 1 on A, u = 0 on ∂Ω \ A.

Conformal Invariance

Under a conformal map f: 𝔻 → Ω with f(0) = z:

$$\omega_\Omega(z, A) = \frac{|f^{-1}(A)|}{2\pi}$$

where |·| denotes arc length on ∂𝔻.

Corner Crowding

For polygonal limits with interior angle α, boundary layers of thickness ρ have:

$$\omega \sim \rho^{\pi/\alpha}$$

For right angles (α = π/2): ω ~ ρ² (quadratic crowding)

API Reference

HarmonicMeasure(geometry)

Main class for harmonic measure computations.

Method	Description
measure_subset(basepoint, subset_fn)	Harmonic measure of subset defined by boolean function
measure_corners(basepoint, radius)	Harmonic measure of corner neighborhoods
density(basepoint)	Harmonic measure density at each boundary node
plot_density(basepoint)	Visualize density on boundary

corner_scaling_diagnostic(curve, ns, c=2.0)

Compute corner harmonic mass across a range of Lamé exponents.

Returns (ns, arc_lengths, fitted_slope).

adaptive_convergence(n, a, c, tol=1e-10)

Adaptively refine discretization until harmonic measure converges.

CLI

# Compute harmonic measure of corner regions
harmonic-measure corners --p 32 --radius 0.1

# Scaling diagnostic across exponents
harmonic-measure scaling --ns 8 16 32 64 --output scaling.csv

# Plot harmonic measure density
harmonic-measure density --p 16 --output density.png

Applications

• Conformal mapping: Diagnose boundary correspondence crowding
• Schwarz-Christoffel limits: Quantify prevertex clustering
• Brownian motion: Exit distribution from domains
• Potential theory: Capacity and extremal length computations

Related Packages

• superellipse: Geometry primitives
• panel-bie: Boundary integral solvers

References

• Garnett, J.B. & Marshall, D.E. (2005). Harmonic Measure. Cambridge University Press.
• Pommerenke, C. (1992). Boundary Behaviour of Conformal Maps. Springer.

License

MIT License. See LICENSE for details.