Determine the leading tail constants for higher Tracy–Widom beta distributions

Determine the leading constants in the tail asymptotics of the higher Tracy–Widom distributions $TW_{\beta}^{(k)}$ for general $\beta>0$ and $k\geq 2$, including the right- and left-tail regimes not resolved by the available results.

Background

The paper studies the higher edge eigenvalues of Hermite and Laguerre β\beta-ensembles and their limiting distributions TWβ(k)TW_{\beta}^{(k)}. Whereas the leading tail constants are known for the largest eigenvalue, the corresponding constants for higher edge eigenvalues are substantially less understood for general β\beta.

The results establish several matching bounds and asymptotics in restricted parameter ranges, but do not determine the complete tail asymptotics for every β>0\beta>0 and k≥2k\geq2. Thus, identifying the missing leading constants remains an unresolved problem beyond the regimes treated in the paper.

References

Even for the limiting distributions $TW_\beta{(k)}$, the leading constants in the tail asymptotics are not known for general $\beta$ and $k\ge2.

— Optimal upper tail estimates for the edge eigenvalues of $β$-ensembles  (2610.11239 - Baslingker, 8 Oct 2026) in Abstract; Section 1, paragraph beginning “However, in sharp contrast with the largest eigenvalue”

It is widely believed that the left tails should not depend on $k$, for all $\beta>0.

— Optimal upper tail estimates for the edge eigenvalues of $β$-ensembles  (2610.11239 - Baslingker, 8 Oct 2026) in Section 1, paragraph immediately preceding Theorem \ref{t: right tail upper bound for beta geq 2}