Category: Physics

Summary: Estimating when a heavy-tailed Wigner matrix plus a four-level diagonal field separates into four balanced spectral bands with three internal gaps.

Random-matrix models often show clean spectral bands only in asymptotic analyses, while finite systems can blur those structures. This experiment asks when a Student-t Wigner matrix deformed by a symmetric four-level diagonal field develops three well-resolved internal gaps, effectively splitting the spectrum into four bands.

The computation builds dense symmetric matrices directly on the GPU and carries a threshold bracket across larger matrix sizes. By tracking when the empirical eigenvalue distribution cleanly separates, it estimates a finite-size splitting threshold for a heavy-tailed deformation problem that blends rare large couplings with multi-level structure.

That is interesting because heavy tails and deformed Wigner ensembles are each well studied, but the finite-size onset of a clean four-band phase is less charted. The experiment aims to quantify how strong the diagonal structure must be before the band picture becomes robust rather than merely suggestive.

Method: GPU dense symmetric eigenspectrum calculation with iterative deepening and bisection on the deformation strength in a Student-t Wigner plus four-level diagonal ensemble.

What is measured: Critical four-band splitting threshold, internal spectral gap formation, band balance, matrix size reached, and bracket width.