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Asymptotic independence of extreme eigenvalues in Gaussian beta-ensembles

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For β∈{1,2,4}\beta\in\{1,2,4\}, let Xβ,nmin⁡X_{\beta,n}^{\min} and Xβ,nmax⁡X_{\beta,n}^{\max} be the centred and rescaled smallest and largest eigenvalues of a random matrix from the Gaussian β\beta-ensemble, defined by

Xβ,nmin⁡:=2n1/6(min⁡Xi+2n),Xβ,nmax⁡:=2n1/6(max⁡Xi−2n).X_{\beta,n}^{\min}:=\sqrt{2}n^{1/6}(\min X_i+\sqrt{2n}),\qquad X_{\beta,n}^{\max}:=\sqrt{2}n^{1/6}(\max X_i-\sqrt{2n}).

Asymptotic independence conjecture. For every x,y∈Rx,y\in\mathbb{R},

lim⁡n→∞(P[Xβ,nmin⁡≤x,Xβ,nmax⁡≤y]−P[Xβ,nmin⁡≤x]P[Xβ,nmax⁡≤y])=0.\lim_{n\to\infty}\Big(\mathbb{P}[X_{\beta,n}^{\min}\le x, X_{\beta,n}^{\max}\le y]-\mathbb{P}[X_{\beta,n}^{\min}\le x]\mathbb{P}[X_{\beta,n}^{\max}\le y]\Big)=0.

This conjecture would provide the joint limit theorem for the smallest and largest eigenvalues needed to analyze the maximum absolute eigenvalue and the critical intersection problem for classical and Schatten pp-balls. The source notes that the assertion is known for β=2\beta=2, while for β∈{1,4}\beta\in\{1,4\} it remains open.

References

Primary source

Mathias Sonnleitner and Christoph Thäle, “A note on critical intersections of classical and Schatten p-balls”, arXiv:2308.10635 (2025).

eigenvalue-statisticsrandom-matrix-theorytracy-widom-distribution
Posted

Progress summary

Refreshed
Open

The two-end behavior is understood in the unitary case, but no public proof has been found for the orthogonal and symplectic cases.

The conjecture asks whether the smallest and largest eigenvalues become independent after edge rescaling. The case β=2\beta=2 is proved, while the cases β=1\beta=1 and β=4\beta=4 remain unresolved.

Known results

  • A 2008 Coulomb-gas study derives joint extreme-eigenvalue asymptotics for β=1,2,4\beta=1,2,4, but does not establish independence for β=1\beta=1 or β=4\beta=4.
  • A 2009 paper proves asymptotic independence for the GUE, hence β=2\beta=2, including an explicit leading correction.
  • A 2011 analytic proof obtains the same factorization for GUE and the Laguerre unitary ensemble, without treating β=1,4\beta=1,4.

Current status (as of August 2026): The conjecture is settled for β=2\beta=2; no verified proof or counterexample is recorded for β=1\beta=1 or β=4\beta=4.

Sources

Solutions 0

No solutions have been posted yet.