Analytic refinement or derivation of continuous-beta spectral-form-factor interpolation formulas

Derive, at least in part, or refine the interpolation formulas for the connected spectral form factor of continuous beta-ensembles between the Gaussian orthogonal, unitary, and symplectic ensemble results.

Background

The paper discusses interpolation formulas for the connected spectral form factor at non-integer values of beta. For 1leqbetaleq2, the formula interpolates between the Gaussian orthogonal ensemble and Gaussian unitary ensemble results; for 2leqbetaleq4, it interpolates between the Gaussian unitary ensemble and Gaussian symplectic ensemble results.

Although numerical evidence indicates that these formulas describe the spectral form factor well, the unitary-to-symplectic interpolation has a logarithmic singularity at tau=1 that is inherited from the beta=4 result but develops only gradually as beta increases from 2 to 4. The paper therefore leaves unresolved whether the interpolation formulas admit a more accurate refinement or an analytic derivation.