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

TL;DR

This paper derives exact gap-ratio results for mixed Wigner surmises involving up to four eigenvalues across different random matrix classes, revealing approximations and coincidences in their distributions.

Contribution

It provides new exact calculations for mixed Wigner surmises with up to four eigenvalues, especially for equal mixtures of GOE, GUE, and GSE classes.

Findings

2xGOE and 2xGUE distributions are well approximated by 2+2 eigenvalue surmises.

2xGSE distribution is better estimated by intermediate statistics between GUE and GSE.

Exact results enhance understanding of eigenvalue spacing in mixed random matrix ensembles.

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.