Baik–Rains interlacing conjecture for GOE and GSE eigenvalues

Let $F_{\beta}(s,m)$ denote the distribution of the $m^{th}$ largest eigenvalue in the appropriate scaling limit for the Gaussian orthogonal ensemble ($\beta=1$) and Gaussian symplectic ensemble ($\beta=4$). Baik–Rains conjecture. In the appropriate scaling limit, the distribution of the largest eigenvalue in GSE corresponds to that of the second largest in GOE. More generally, the joint distribution of every second eigenvalue in the GOE coincides with the joint distribution of all the eigenvalues in the GSE, with an appropriate number of eigenvalues. This conjecture concerns the interlacing relationship between eigenvalue distributions in the orthogonal and symplectic ensembles; its verification is stated in the source to be the content of a corollary, so the parser's unresolved status is retained here pending explicit resolution evidence.