Exact gap-ratio results for mixed Wigner surmises of up to 4 eigenvalues

Abstract

We compute some exact results for the gap-ratio of mixed Wigner surmises for up to four eigenvalues and $0\leq\beta\leq4$. The main results concern equal mixtures of the GOE, GUE, and GSE random matrix classes. These give rise to $2\times$GOE, $2\times$GUE, and $2\times$GSE~distributions. We find that $2\times$GOE, $2\times$GUE are well approximated by the surmises of only 2+2 eigenvalues that are GOE and GUE distributed, respectively. The same is not valid for $2\times$GSE, which is well estimated, by coincidence, by 2+2 eigenvalues of statistics intermediate between GUE and GSE.