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Baik–Rains interlacing conjecture for GOE and GSE eigenvalues

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Let Fβ(s,m)F_{\beta}(s,m) denote the distribution of the mthm^{th} largest eigenvalue in the appropriate scaling limit for the Gaussian orthogonal ensemble (β=1\beta=1) and Gaussian symplectic ensemble (β=4\beta=4). Baik–Rains conjecture. In the appropriate scaling limit, the distribution of the largest eigenvalue in GSE corresponds to that of the second largest in GOE. More generally, the joint distribution of every second eigenvalue in the GOE coincides with the joint distribution of all the eigenvalues in the GSE, with an appropriate number of eigenvalues. This conjecture concerns the interlacing relationship between eigenvalue distributions in the orthogonal and symplectic ensembles; its verification is stated in the source to be the content of a corollary, so the parser's unresolved status is retained here pending explicit resolution evidence.

References

Primary source

Momar Dieng, “Distribution Functions for Edge Eigenvalues in Orthogonal and Symplectic Ensembles: Painlevé Representations”, arXiv:math/0411421 (2004).

eigenvalue-distributionsorthogonal-symplectic-ensemblesrandom-matrix-theory
Posted

Progress summary

Refreshed
Claimed solved

The conjecture was proved in 2005: every other limiting eigenvalue in the orthogonal ensemble has the corresponding distribution in the symplectic ensemble.

Baik and Rains posed the conjecture in 2001, asserting an interlacing correspondence between GOE and GSE eigenvalue distributions. The central identity is

F4(s,m)=F1(s,2m),m≥1.F_{4}(s,m)=F_{1}(s,2m),\qquad m\ge 1.

Known results

  • Forrester and Rains proved the corresponding equivalence between alternate GOE eigenvalues and GSE eigenvalues at finite ensemble size.

2005 proof

The 2005 paper proves the full interlacing statement, including the joint distribution of every second GOE eigenvalue and the corresponding GSE eigenvalues. Its Corollary 2.2 verifies the Baik–Rains conjecture and gives the displayed identity; it also corrects the related distinction that the conjecture holds for D4(s,λ)D_{4}(s,\lambda) but not for D1(s,λ)D_{1}(s,\lambda).

Current status (as of August 2026): the Baik–Rains interlacing conjecture is proved, including its joint-distribution form and F4(s,m)=F1(s,2m)F_{4}(s,m)=F_{1}(s,2m) for m≥1m\ge 1, with no unresolved objection recorded.

Sources

Solutions 0

No solutions have been posted yet.