Category: Statistics

Summary: Estimating when a smooth rank-one signal becomes recoverable inside heavy-tailed random noise with Toeplitz-profile correlations.

Spiked random-matrix models ask when a weak structured signal can be separated from noise. This experiment studies that question in a harder setting where the additive noise is both heavy tailed and correlated across distance through a Toeplitz variance profile, then asks for the finite-size threshold at which a smooth rank-one spike becomes reliably detectable.

The script builds dense symmetric matrices, inserts a smooth spike, and uses iterative deepening with repeated bisection over spike strength while increasing matrix size. Detection is based on spectral signatures such as eigenvector overlap and gap behavior, so the output is a threshold map rather than a single scan.

That combination of heavy tails, structured correlations, and finite-size threshold tracking is the main scientific target. It tests how much classical BBP-style intuition survives once the background noise is no longer Gaussian or spatially uniform.

Method: Dense symmetric eigensolves with iterative deepening and bisection on spike strength in heavy-tailed Toeplitz-profile random matrices.

What is measured: Critical spike threshold, eigenvector overlap, spectral gap ratio, finite-size bracket width, and system size reached.