Entrywise Eigenvector Fluctuations

• Entrywise eigenvector fluctuations are a key concept in random matrix theory, describing the probabilistic behavior of individual eigenvector coordinates in spiked models.
• They employ resolvent methods and CLT-type results to quantify fluctuations, demonstrating universality in delocalized regimes and sensitivity in localized settings.
• These insights guide spectral methods in PCA and signal processing by delineating phase transitions, alignment mechanisms, and error rates in high-dimensional inference.

Entrywise eigenvector fluctuations concern the probabilistic behavior of individual coordinates of eigenvectors of large random matrices, especially in the presence of low-rank perturbations ("spikes") and in regimes where nontrivial alignment with signal directions emerges. This topic is central in random matrix theory, high-dimensional statistics, and signal processing, particularly for spectral methods applied to problems such as principal component analysis, detection and estimation in noisy environments, and inference in spiked models. Entrywise fluctuation theory seeks to rigorously describe, quantify, and classify the distributional limits, universality, and typical size of eigenvector entries, with critical attention to the delocalization-to-localization phase transition and the functional dependence on model specifics.

1. Model Framework and Definition of Entrywise Fluctuations

Consider a rank-one spiked matrix model of the form

$H = \theta vv^* + W$

where $H\in\mathbb{C}^{n\times n}$ is Hermitian, $\theta\in\mathbb{R}$ is the spike strength, $v\in S^{n-1}(\mathbb{C})$ is a deterministic or random (delocalized) unit-norm signal, and $W$ is a (generalized) Wigner noise matrix with independent entries, bounded variance profile, and sub-exponential tails.

Let $\widehat{v}$ denote the top (normalized) eigenvector of $H$. Entrywise eigenvector fluctuations refer to the joint and marginal distribution of

$\big\{ \widehat{v}_i : 1\leq i\leq n \big\}$

in the large $n$ limit, with centering and scaling that may depend on the signal-to-noise ratio (SNR), the underlying spike, and the noise law. The supercritical regime $\theta>1$ is of special interest: the principal eigenvalue exits the spectral bulk and $\widehat{v}$ aligns with $v$, but nontrivial random fluctuations around the aligned direction are observed entrywise (Chen et al., 12 Dec 2025).

2. Universality and Limiting Laws in the Delocalized Regime

In the case of sufficiently delocalized spikes,

$$\|v\|_\infty \leq C_v n^{-1/2+\alpha}, \qquad \alpha\in(0,1/20)$$

the entrywise distribution of $\widehat{v}$ exhibits universality: for a large class of generalized Wigner ensembles $W$, the limiting law of the entries depends only on the global spike strength $\theta$ and the first two moments of the noise entries, but not on higher moments (Chen et al., 12 Dec 2025).

In this regime, for any fixed $i$,

$$\sqrt{n}\left[\widehat{v}_i - \rho(\theta)\,v_i\right] \xrightarrow{d} \mathcal{N}(0,\tau^2(\theta))$$

with $\rho(\theta) = \sqrt{1-\theta^{-2}}$, $\tau^2(\theta) = \theta^{-2}$. This CLT-type result remains valid for any function of entries or averaged entrywise statistics, as long as the centering and variance properly capture the model parameters. The entrywise universality theorem is robust to the choice of $W$ as long as the variance profile is flat (up to $n^{-1-o(1)}$ corrections) and the delocalization condition on $v$ holds (Chen et al., 12 Dec 2025).

This universality is lost for localized spikes: the distribution of entries and the top eigenvalue itself can then be sensitive to higher moments of $W$ (Chen et al., 12 Dec 2025).

3. Analytical Techniques and Key Theorems

The analytical approach to entrywise eigenvector fluctuations combines local laws for the resolvent $(H-z)^{-1}$, Taylor/Lindeberg replacement methods, and Riesz/projection formulae for characterizing eigenvector entries. The central technical object is the resolvent-based representation: $$n\,\widehat{v}_i\,\overline{\widehat{v}_j} \approx \frac{(e_i^* R_W(\lambda) v)\,(v^* R_W(\lambda) e_j)}{v^* R_W(\lambda)^2 v}$$ evaluated at the outlier eigenvalue $\lambda = \theta + \theta^{-1}$ (Chen et al., 12 Dec 2025). Moment matching (Gaussian comparison) arguments are then used to establish that, under matching covariance structures, the entrywise statistics match those for the corresponding Gaussian case up to $n^{-1/2+o(1)}$.

In the case of nonlinear spiked matrix models, the emergence of entrywise (generalized) eigenvector fluctuations is controlled by the hierarchy of generalized information coefficients, specifically the first non-vanishing coefficient $\vartheta_{k_{\star}}(f)$ and the associated phase transition scaling of the spike (Guionnet et al., 2023). After reduction to an effective rank-one model, fluctuation results for the top eigenvector's alignment (and hence its entries) can be obtained by invoking the classical BBP theory and its local expansions (Guionnet et al., 2023).

For spiked separable covariance and Haar-invariant multiplicative deformations, anisotropic local laws provide the entrywise control required for both in-bulk (delocalized) and outlying (localized) eigenvectors, with explicit cone-concentration and scaling asymptotics (Ding et al., 2019, Ding et al., 2023).

4. Delocalization, Localization, and Phase Transitions

A key phenomenon is the phase transition at the BBP threshold (e.g., $\theta=1$ for standard Wigner): below this critical value, $\widehat{v}$ is fully delocalized in the ambient basis,

$|\widehat{v}_i|^2 \asymp n^{-1}$

and its entrywise fluctuations are asymptotically trivial, with vanishing correlation to $v$ (Chen et al., 12 Dec 2025, Perry et al., 2018, Miolane, 2018). Above the threshold, the top eigenvector aligns with the spike and exhibits nontrivial entrywise fluctuations as described above.

For outlier eigenvectors in spiked multi-spike or separable models, the cone-concentration phenomenon precisely quantifies the deterministic alignment and the aperture of fluctuations around the nominal direction, with explicit dependence on population-level spike strengths and Stieltjes transforms (Ding et al., 2019, Ding et al., 2023). Non-outlier ("bulk") eigenvectors remain fully delocalized under suitable regularity (Ding et al., 2019, Ding et al., 2023).

In models with diverging spikes and appropriate delocalization conditions, both the eigenvalues and the entries of eigenvectors satisfy explicit asymptotic expansions and finite-sample CLTs, with precise variance and bias terms depending on the model's structural parameters (Fan et al., 2019).

5. Fluctuation Theory Under Nonlinear and Non-Gaussian Models

For entrywise nonlinear models (e.g., when a function $f$ is applied entrywise to a spiked Wigner matrix), the spectral properties and entrywise eigenvector fluctuations are controlled by the generalized information coefficients $\vartheta_k(f)$ (Guionnet et al., 2023). The phase transition scaling of the spike strength, critical for nontrivial entrywise behavior, is dictated by the smallest index $k_{\star}$ with $\vartheta_{k_{\star}}(f) \neq 0$. Post-transition, the top eigenvector aligns not with $x$, but with $x^{\odot k_{\star}}$, and entrywise fluctuations reflect this change of basis (Guionnet et al., 2023).

In non-Gaussian Wigner models, it is proven that principal component analysis (PCA) is information-theoretically suboptimal, and the entrywise application of a nonlinearity (score function) $f(w) = -p'(w)/p(w)$ restores both the detection threshold and the optimality of PCA-like methods for the top eigenvector (Perry et al., 2018).

6. Applications: Spectral Algorithms and Statistical Inference

Entrywise fluctuation results underlie the analysis of rounded spectral algorithms in problems such as the stochastic block model (SBM) and group synchronization. Precise characterizations of asymptotic error rates and label recovery thresholds are derived by propagating the entrywise fluctuation law of the top eigenvector through the rounding (e.g., sign or phase) nonlinearity, leading to "single-letter" loss formulas in the $n\to\infty$ limit (Chen et al., 12 Dec 2025). These analyses illuminate the close connection between spectral phase transitions and algorithmic performance in high-dimensional inference.

Detectability and inference limits are tied directly to the scaling of entrywise fluctuations and the transition between delocalized and aligned regimes. When spikes accumulate into the spectral bulk or their mass is spread too thinly, no localization or persistent outlier eigenvectors remain, and all eigenvectors become delocalized at the $O(n^{-1/2})$ scale (Thompson, 2019, Miolane, 2018).

7. Summary Table: Entrywise Eigenvector Fluctuation Laws (Delocalized Regime)

Model Class	Entrywise Limiting Law	Universality
Spiked Wigner (delocalized $v$)	$\mathcal{N}(\rho(\theta)v_i,\,\theta^{-2}/n)$	First two moments of $W$ determine the limit (Chen et al., 12 Dec 2025)
Nonlinear spike model	As above for effective spike $x^{\odot k_\star}$	Dictated by $\vartheta_{k_\star}(f)$ (Guionnet et al., 2023)
Separable covariance, Haar-invariant	Explicit cone, controlled aperture (Ding et al., 2019, Ding et al., 2023)	Holds under spectral regularity/polynomial moments
Diverging spike (generalized Wigner)	CLT with explicit bias/variance (Fan et al., 2019)	Closed-form in resolvent series

The universality statements are restricted to delocalized spike regimes; localization of $v$ breaks universality due to higher moment effects.

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The study of entrywise eigenvector fluctuations thus reveals not only detailed probabilistic laws for principal components in spiked and deformed random matrix models but also provides rigorous justification and performance guarantees for spectral inference and estimation procedures in high dimensions (Chen et al., 12 Dec 2025, Guionnet et al., 2023, Perry et al., 2018, Fan et al., 2019, Ding et al., 2023, Ding et al., 2019, Miolane, 2018, Thompson, 2019).