Analytical second-order spacing-ratio crossover distribution

Derive an analytical expression for the intermediate statistics of the second-order spacing ratio ($k=2$) in the GOE--GUE crossover described by the Pandey--Mehta random-matrix model.

Background

The paper studies first- and second-order spacing-ratio distributions as diagnostics of the GOE--GUE crossover in Hermitian adjacency matrices of random mixed graphs. An exact interpolating distribution is available for the consecutive spacing ratio ($k=1$), but the analogous intermediate distribution for the second-order ratio ($k=2$) is not available analytically. Consequently, the second-order crossover is treated numerically rather than through a closed-form theoretical benchmark.