Exact calculation of the distribution of every second eigenvalue in classical random matrix ensembles with orthogonal symmetry

The explicit quaternion determinant formula for the n-point distribution of the even numbered eigenvalues (ordered so that x1 < x2 < · · ·) in the classical random matrix ensembles with orthogonal symmetry is computed. For an odd number of eigenvalues N +1 it is found to coincide with the n-point distribution for the eigenvalues in the corresponding ensemble with symplectic symmetry and N/2 eigenvalues, although in the Laguerre and Jacobi cases the parameters must be modified. In the Gaussian case our result says that the joint distribution of every second eigenvalue in the GOE with an odd number of eigenvalues N + 1 coincides with the joint distribution of all the eigenvalues in the GSE with N/2 eigenvalues, in agreement with a recent conjecture of Baik and Rains. Also verified is another conjecture of Baik and Rains, which relates the distribution of two superimposed GOE spectra, with each second eigenvalue integrated out, to the distribution of the GUE. Here equality is restricted to the large N limit.