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 ≤ β ≤ 4. The main results concern equal mixtures of the GOE, GUE, and GSE random matrix classes. These give rise to 2×GOE, 2×GUE, and 2×GSE distributions. We find that 2×GOE, 2×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×GSE, which is well estimated, by coincidence, by 2+2 eigenvalues of statistics intermediate between GUE and GSE.

Relaxation mechanisms in a disordered system with the Poisson level statistics

“…In the vicinity of the transition, P (r) can be well approximated by a mixed Wigner surmise discussed very recently in Ref. [59]. More precisely, the dashed curve in Fig.…”

“…while an approximate distribution for the Gaussian-Orthogonal-Ensemble (GOE) can be derived from the Wigner surmise [57][58][59] for 3 energy levels…”

“…( 20) in Ref. [59] with a normalization that is relevant for r ∈ (0, 1). 6 but for a full spectra containing symmetric and antisymmetric levels.…”

“…( 24) in Ref. [59] with a normalization that is relevant for r ∈ (0, 1), whereas the dashed line in (b) shows Eq. (12).…”

Bipartite Sachdev-Ye-Kitaev model: Conformal limit and level statistics

“…As discrete symmetries are common in quantum Hamiltonians, spectra formed out of two or more independent GOE or GUE components are the subject of longstanding interest in the quantum-chaos and random-matrix literature [35,[47][48][49][50][51][52][53][54][55][56][57][58][59][60]. Here, we will only be concerned with the 2 × GOE case because restricting the couplings of the SYK Hamiltonian to obtain the b-SYK Hamiltonian effectively adds a single Z 2 symmetry.…”