Sensor-averaged eigenfunction independence underlying Berry isotropy

Prove or disprove that, for generic sensor placements drawn uniformly from a bounded two-dimensional domain, distinct Laplacian eigenfunctions have approximately uncorrelated sampled values, specifically that the cross-correlations C_kn=(1/M)∑_{m=1}^M φ_k(x_m)φ_n(x_m) satisfy E[C_kn^2]≈1/M for k≠n under the stated sensor normalization.

Background

The paper uses a sensor-level version of Berry’s random-wave conjecture to explain why truncation-noise covariance is approximately isotropic. If high-frequency eigenfunctions behave as sufficiently decorrelated random fields at uniformly sampled microphone locations, the discarded-mode contributions should average toward an isotropic covariance, supporting the closed-form diagonal regularizer Γ_k∝λ_k^s.

The paper reports empirical validation of the conjecture in simulated rooms, but the statement remains a conjecture rather than a proved result for the generic bounded domains and sensor placements considered. Its validity is important because the subsequent anisotropy bounds and regularizer arguments rely on this decorrelation mechanism.