Category: Physics

Summary: Mapping finite-size corrections to the bulk-splitting transition in a Wigner matrix plus a deterministic two-block deformation.

Free probability predicts that adding a two-block deterministic diagonal matrix to a GOE matrix causes the spectral bulk to split once the deformation strength reaches a critical value. This experiment asks how that transition looks at finite size, where the asymptotic threshold is known but the correction law and edge statistics near the transition are not well characterized.

The model generates symmetric random matrices of the form Wigner plus deterministic two-block structure and then bisects the deformation strength to estimate the apparent critical value delta_c(N). Repeating the procedure across many seeds and matrix sizes produces a map of how the transition drifts toward its asymptotic limit.

That makes the project a finite-size free-probability study rather than a check of the asymptotic theorem itself. The value lies in quantifying the approach to the known critical point and probing what kind of spectral statistics emerge right at the crossover.

Method: GPU symmetric eigensolves with bisection on deformation strength delta for matrices of the form GOE plus a deterministic two-block diagonal perturbation.

What is measured: Apparent critical delta_c(N), finite-size drift, bulk-splitting crossover, and transition-region spectral statistics.