Category: Physics

Summary: Estimating the bandwidth prefactor where random band matrices cross from Wigner-Dyson to Poisson spectral statistics.

Random band matrices sit between fully connected random matrices and strictly local systems, and they are a standard toy model for localization in disordered media. Their eigenvalue spacings change from level-repelling Wigner-Dyson behavior to Poisson behavior as the band narrows. Theory predicts the critical bandwidth should scale like the square root of system size, but the numerical prefactor is still not known in closed form.

This experiment generates symmetric band matrices, measures edge-spacing behavior, and bisects the rescaled bandwidth parameter alpha = b/sqrt(N). The iterative-deepening design carries a narrowing threshold bracket to larger matrix sizes, building a direct estimate of the transition point rather than scanning the whole parameter range from scratch at every size.

That produces a dense finite-size map of the crossover relevant to Anderson localization, quasi-one-dimensional transport, and quantum chaos. The value lies in resolving the transition systematically across large N rather than relying on sparse legacy sweeps.

Method: Dense symmetric eigensolve with iterative deepening and bisection on the rescaled bandwidth alpha = b/sqrt(N).

What is measured: Critical alpha, implied critical bandwidth, bracket width, passes completed, and spacing-statistics references for GOE and Poisson behavior.