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.
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.
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.
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).
What is left to future work is the exact location of the noise-bulk upper edge: an edge-rigidity formalization of the MP edge $\sigma_{\mathrm{noise}}2(1+\sqrt{\gamma})2$, the BBP threshold \citep{bbp}, and the Tracy--Widom fluctuation \citep{tw} under short-range mixing. The state of the citation chain: the quadratic-form concentration criterion of Bai--Zhou \citep{baizhou} and its verifications under m-dependence (Hui--Pan \citep{huipan}) and for linear processes (Yao ; Pfaffel--Schlemm \citep{pfaffel}) give the limiting spectral distribution under mixing (the bulk shape); Bai--Silverstein \citep{baisilv} give the no-eigenvalue-outside-the-support conclusion for independent columns; El Karoui \citep{elkaroui} gives the Tracy--Widom limit for Toeplitz-type population covariances; the missing piece is the last step carrying ``no eigenvalues outside the support + edge fluctuation'' over to short-range-mixing sample covariances.