width-null-energy-random-matrix-theory
IN premise — summaries/2026/08/24/aristotelian-2026-sR-references.md
Created 2026-08-24T17:10:51+00:00
Under H₀ (independence), the expected squared Frobenius norm of the sample cross-covariance is E[‖Ĉ‖²_F] = dₓd_y/(n−1), so when dₓ and d_y are comparable to n, the null energy is O(n) and does not vanish as n→∞
Summary
When the number of variables in each set is on the order of the sample size, the pure-noise "energy" in the cross-covariance matrix scales linearly with n rather than shrinking toward zero, so adding more data does not clean up the noise floor. In practice, this means any detector trying to find a real relationship between two variable sets must beat a noise level that grows with the sample, making the high-dimensional detection problem fundamentally harder than the classical low-dimensional case.