Category: Physics

Summary: Estimating how much mesoscopic block correlation in heavy-tailed Wigner noise is needed before a planted spike reliably realigns the top eigenvector.

Spiked random-matrix models ask when a hidden signal becomes visible against noise, but most standard results assume simpler noise than real structured systems contain. This experiment studies a planted mixed-scale spike inside a dense symmetric matrix whose background noise is both heavy tailed and block correlated, then asks for the correlation strength at which the dominant eigenvector starts to lock onto the hidden signal.

The script generates Student-t Wigner matrices with mesoscopic block structure, adds a planted spike, and uses iterative deepening to bisect the block-correlation parameter across increasing matrix sizes. At each pass it measures whether the leading eigenvector aligns with the spike strongly enough to count as recovery, together with eigenvalue separation and localization diagnostics.

That makes the project a finite-size threshold map for structured spike recovery rather than a generic random-matrix simulation. The interest is in how rare large entries and block organization combine to shift the signal-detection boundary away from the cleaner BBP-style setting.

Method: Dense symmetric eigensolves with iterative deepening and bisection on block-correlation strength rho for heavy-tailed spiked Wigner matrices from N=64 to 2048.

What is measured: Critical block-correlation threshold, spike-recovery fraction, top-eigenvector alignment, top and runner-up eigenvalues, localization diagnostics, and bracket width.