Category: Statistics

Summary: Finding when heavy-tailed random matrices develop a spectral gap under a binary diagonal shift.

In free probability theory, adding a deterministic diagonal matrix to a random Wigner matrix can split the eigenvalue distribution into separate clusters (a spectral gap). This is well-understood for Gaussian entries, but what happens with heavy-tailed Student-t entries?

This experiment maps the critical diagonal shift strength where a Student-t Wigner matrix with a two-point diagonal potential develops a clean central spectral gap. Heavy tails create stronger outlier eigenvalues that may resist or enhance the gap formation.

This bridges deformed Wigner matrix theory with heavy-tailed random matrix universality in a way that has not been directly studied at finite matrix sizes.

Method: GPU dense eigensolve (cupy.linalg.eigvalsh) with iterative deepening. Bisection on diagonal shift strength.

What is measured: Critical shift for gap opening, gap width, bulk edge positions, outlier statistics.