Tracy–Widom edge universality for power-law correlations above the flatness threshold

Establish whether the row-independent power-law ensemble with covariance \(d_t=(1+t)^{-\gamma}\) exhibits Tracy–Widom edge universality for \(\gamma>1\), including verification of the boundedness assumption for the self-consistent Matrix Dyson Equation solution.

Background

For γ>1\gamma>1, the power spectral density is bounded and the flatness assumption required by the Matrix Dyson Equation approach holds. However, the boundedness of the self-consistent solution near the edge, Assumption (G), is not automatic from the other hypotheses.

Consequently, the paper treats Tracy–Widom behavior in this regime as conjectural rather than as a theorem.

References

We therefore regard TW edge universality for \gamma>1 as a conjecture on the same footing as Conjecture~\ref{conj:TW} for the AR(1) case, not as a consequence of Proposition~\ref{prop:two-thresholds}(ii) alone.

— Bulk Phase Transition and Edge Behavior in Temporally Correlated Random Matrices  (2608.23944 - Hisakado et al., 25 Aug 2026) in Remark~\ref{rem:logical-status}, Section IV.D

Both the existence of such a root for 1<\gamma\le3/2, and the behavior of \tau_0(\gamma) as \gamma\to1+, are left as open questions.

— Bulk Phase Transition and Edge Behavior in Temporally Correlated Random Matrices  (2608.23944 - Hisakado et al., 25 Aug 2026) in Remark~\ref{rem:log-correction} [actually labelled Remark: The divergence as \(\gamma\to1^+\) is not established], Section IV.F

We were not able to determine numerically whether this crossover constitutes a genuine non-analyticity in the \gamma-dependence of the edge behavior (a phase transition, in the sense that s_\infty(\gamma) has a singular approach to zero) or a smooth crossover that merely happens to sharpen near \gamma=1 because of the analytic threshold of Assumption~(E).

— Bulk Phase Transition and Edge Behavior in Temporally Correlated Random Matrices  (2608.23944 - Hisakado et al., 25 Aug 2026) in Section IV.G, immediately following Observation~\ref{obs:binder}