Category: Physics

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

Random-matrix theory can often predict when deformations should split a spectrum in the infinite-size limit, but those asymptotic statements do not directly tell us where a finite computation will actually show clean band separation. This experiment asks when a Student-t Wigner matrix, perturbed by a symmetric four-level diagonal field, develops three internal gaps that visibly divide the spectrum into four balanced bands.

The script increases matrix size through iterative deepening and repeatedly bisects the diagonal-field strength. At each stage it performs dense symmetric eigensolves and carries forward only the narrowing bracket around the apparent splitting point, so the computation focuses on the transition itself instead of scanning the full parameter range from scratch.

That makes the project a finite-size threshold map for a structured heavy-tailed deformation problem. The interest is in resolving when clean multi-band structure emerges numerically, not just whether free-convolution theory predicts that splitting in principle.

Method: Dense symmetric eigensolves with iterative deepening and bisection on the four-level diagonal-field strength in a Student-t Wigner ensemble.

What is measured: Critical splitting strength, internal gap formation, band balance, system size reached, and threshold bracket width.