LAUSR

The joint distribution of every second eigenvalue obtained after superposing the spectra of two circular orthogonal ensembles ($\beta=1$) is known to be equal to that of the circular unitary ensemble ($\beta=2$), where the parameter $\beta$ is the Dyson index of the ensemble. Superposition of spectra of $m$ such circular orthogonal ensembles is studied numerically using higher-order spacing ratios. It is conjectured that the joint probability distribution of every $k=m-2$-th $(m\geq4)$ eigenvalue corresponds to that of circular $\beta$-ensemble with $\beta=m-3$. For the special case of $m=3$, $k=\beta=3$. It is also conjectured that the spectral fluctuations corresponding to $k=m+1$ ($m\geq2$) and $k=m-3$-th ($m\geq5$) order spacing ratio distribution is identical to that of nearest neighbor spacing ratio distribution with Dyson indices $m+2$ and $m-4$ respectively. Strong numerical evidence in support of these conjectures is presented.