Extend stochastic domination of top-eigenvalue sums to individual higher eigenvalues

Prove that the stochastic comparison in Proposition \ref{prop: sum of top k stoch dom} extends from sums of the top $k$ eigenvalues to the $k$-th largest eigenvalue itself across Hermite and Laguerre $\beta$-ensembles with the parameter relations specified there.

Background

The proposition establishes stochastic domination for sums of the top kk eigenvalues when the inverse-temperature parameters satisfy a rationality relation. Such a comparison is useful for transferring tail estimates between different β\beta-ensembles.

The author states that the analogous comparison for the individual kk-th largest eigenvalue is believed to hold but is not proved. Establishing this extension would strengthen the available finite-dimensional comparison theory for higher edge eigenvalues.

References

Although we do not have a proof, we believe that Proposition \ref{prop: sum of top k stoch dom} holds for $k$-th largest eigenvalues as well.

— Optimal upper tail estimates for the edge eigenvalues of $β$-ensembles  (2610.11239 - Baslingker, 8 Oct 2026) in Section 1, immediately after Proposition \ref{prop: sum of top k stoch dom}