Category: Physics

Summary: Estimating the spike strength needed for a detectable outlier eigenvalue when a sparse random matrix has both dilution and heavy-tailed noise.

In spiked random-matrix problems, a sufficiently strong signal can peel away from the noisy bulk and become spectrally detectable. This experiment asks how that detectability threshold changes when the background matrix is both sparse and heavy-tailed, combining two ways that classical Gaussian theory can fail.

The model fixes a dilution regime, draws symmetric heavy-tailed noise, adds a rank-one spike, and then uses iterative deepening with bisection to find the critical spike strength where the top eigenvalue separates more than half the time. Increasing system size turns the output into a finite-size threshold map rather than a single-size estimate.

That interaction between sparsity and heavy tails is not captured by the standard dense BBP picture. The experiment is designed to show how rare large entries and missing couplings jointly reshape spectral signal detection.

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

What is measured: Critical spike strength, outlier-detection probability, top-eigenvalue separation, system size reached, and bracket width.