Origin of oscillatory behavior in the fitted higher-dimensional spacing parameter

Explain why the fitted Gaussian-beta parameter and mean normalized spacing ratio for the scattering-angle curves in the high-excited-string-state scattering process oscillate with the excitation level, whereas the corresponding quantities for the polarization-related curves vary monotonically.

Background

The paper analyzes ratios of adjacent areas between non-intersecting curves defined by zeros of derivatives of a high-excited bosonic string scattering amplitude. The distributions are fitted to Gaussian-beta ensemble formulas for two curve families associated with the scattering angle and polarization angle.

The fitted parameters approach random-matrix-like values at larger excitation levels, but the two curve families behave differently: the polarization-related quantities vary monotonically while the scattering-angle quantities oscillate. The paper explicitly identifies the reason for this contrast as unresolved.

References

We can also observe that whereas $\beta_{(y)}$ and $\langle\tilde r_{(y)}\rangle$ are monotonically decreasing their analogs in $x$ oscillate. It is not clear why that is the case.

— Statistical Characterization of Erratic and Chaotic Processes through Their Extrema I. Thermodynamics  (2610.21620 - Sonnenschein, 15 Oct 2026) in Section String scattering HES+T→T+T, discussion following Figure area spacings