Spiked matrix model eigenvector fluctuation conjecture

Let $\theta>1$, let $d_0\geq 1$ be fixed, and use the spiked matrix model notation of Theorem~, assuming and. Let $\bm R=\bm R^{(d)}\in\mathbb{R}^{d_0\times d}$ satisfy $\|\bm R\|=O(1)$, $\bm R\bm v=\bm R^{(d)}\bm v^{(d)}=\bm 0$, and

$$d\cdot\bm R\bm\Sigma\bm R^{\top}\to\bm\Gamma\in\mathbb{R}^{d_0\times d_0}_{\operatorname{sym}},$$

where

$$\bm\Sigma=\bm S^{\star}-(1-\theta^{-2})\bm v\bm v^{\top}.$$

Assume that the dual Lehner formula at $\bm A_0=\theta\bm v\bm v^{\top}$ has a unique optimizer $\bm S^{\star}$ for every $d\geq 1$, and choose $\widehat{\bm v}$ so that $\langle\widehat{\bm v},\bm v\rangle>0$. Spiked matrix model eigenvector fluctuation conjecture. As $d\to\infty$,

$$\mathsf{Law}\left(\sqrt{d}\,\bm R\left(\widehat{\bm v}-\sqrt{1-\theta^{-2}}\,\bm v\right)\right)\to\mathcal{N}(\bm 0,\bm\Gamma).$$

The same conclusion with the same limit should hold when $\bm\Sigma$ is replaced by

$$\bm\Sigma^{\mathsf{est}}=\bm S^{\mathsf{est}}-(1-\theta^{-2})\bm v\bm v^{\top},$$

where $\bm S^{\mathsf{est}}$ is from Definition~, provided Proposition~ makes it well-defined. This conjecture proposes Gaussian projected fluctuations for the leading eigenvector, extending the first-order BBP eigenvector limit; its status is unresolved in the supplied source.