Category: Statistics

Summary: Estimating the spike strength needed to detect a structured signal in heavy-tailed random band matrices near the localization crossover.

Random-matrix spike detection is well studied for broad dense ensembles, while finite-bandwidth matrices are a standard model for localization. This experiment combines those settings and asks how strong a planted spike must be before it separates from heavy-tailed band disorder strongly enough to become detectable.

The script generates dense symmetric band matrices with Student-t distributed entries, adds a low-rank deformation, and then bisects the spike strength while increasing matrix size. By focusing computation on the narrowing transition zone, the experiment maps a BBP-like detection threshold in a regime where heavy tails and quasi-one-dimensional band structure both matter.

That combination is scientifically interesting because rare large entries and limited bandwidth can each reshape eigenvectors and spectral separation. The experiment tests how those two effects interact instead of treating them as independent corrections to a standard spiked model.

Method: Dense symmetric eigensolve with iterative deepening and bisection on spike strength in heavy-tailed band random matrices.

What is measured: Critical spike threshold, spectral separation behavior, implied detection crossover, and threshold bracket width.