The Eigenvector Bead Process

TL;DR

The paper investigates how eigenvectors of a random matrix and its principal minor relate under microscopic growth, for both Wigner and Wishart ensembles. It develops secular/master equations for eigenvalues and eigenvectors under rank-one updates and analyzes their edge and bulk asymptotics, revealing a local projection in the bulk and a small random antisymmetric rotation of order $N^{-1/3}$ at the edge, governed by Airy and Sine kernel universality. It also demonstrates universality across ensembles by showing Wishart matrices exhibit the same bead-type eigenvector dynamics, with edge laws governed by Gamma and Bessel processes and bulk laws by sine$_{\beta}$ statistics, up to $q$-dependent constants. These results provide a universal, quantitative description of eigenvector evolution under minor enlargement, with implications for stability of eigenbases in large random systems and connections to well-known universality classes in random matrix theory.

Abstract

We investigate the overlap matrix between the eigenvectors of a Wigner matrix $H_{N+K}$ of size $(N+K)\times(N+K)$ and those of its principal minor $H_N$ of size $N\times N$, for both the real symmetric ($β=1$) and complex Hermitian ($β=2$) ensembles, in the regime where $N \to \infty$ while $K$ remains fixed. Our analysis yields two main results. (i) In the \emph{bulk} of the spectrum, an eigenvector of $H_{N+K}$ associated with an eigenvalue at energy level $E$ projects primarily onto eigenvectors of $H_N$ located at the same local spectral level. This phenomenon, which we call \emph{local projection}, highlights a robust stability of the eigenbasis under matrix growth. (ii) At the \emph{spectral edge}, the change of basis between the leading eigenspaces of consecutive minors is asymptotically governed by a random antisymmetric perturbation of order $N^{-1/3}$. In both cases, we provide the asymptotic law of the overlaps expressed in terms of the Airy and Sine kernels. We further extend our analysis to the case of Wishart matrices, that is, sample covariance matrices of the form $W = X^{\!\top} X$, where $X \in \mathbb{R}^{T \times N}$ is a matrix with i.i.d.\ random entries. We establish analogous results for the overlaps between eigenvectors of consecutive minors of $W$, both in the bulk and at the spectral edges (soft and hard). The limiting laws share the same universal structure as in the Wigner case, up to explicit constants depending on the aspect ratio $q = N/T$. This demonstrates the universality of the eigenvector overlap process across distinct random matrix ensembles.

Figures (4)

• Figure 1: Mean absolute overlap as a function of the offset $k=i-j$. Panels (a)–(b) plot $\mathbb{E}\left[|\Omega^{(N)}_{i,i-k}|\right]$ for $N=100$ at energies (a) $E=1.4$ and (b) $E=0.3$. Blue circles show the empirical averages: in each run we draw a real-symmetric Wigner matrix ($\beta=1$), augment it by one row and one column, and then compute the overlap between the relevant eigenvectors. Red dashed squares show the theoretical simulation: we sample the sine--$\beta$ process at $\beta=1$ with Gaussian variables, and evolve the inserted level via the secular equation $S(z)=h$. Evaluating the derivative $S'(z)$ at the corresponding root yields the predicted overlaps according to formula \ref{['eq:bulk-near-anchored']}. Results are averaged over $10{,}000$ Monte Carlo runs.
• Figure 2: Distribution of absolute overlaps at offsets $k\in\{-1,0,1,2\}$. Panels compare the empirical (numerical, blue) and predicted (theoretical, red) densities of $|\Omega^{(N)}_{i,\,i-k}|$ for $N=100$ at energy $E=0.3$. We fix the reference index $i=60$ and examine the four neighbour indices $j=i+k\in\{58,59,60,61\}$ (i.e., $k=-1,0,1,2$). Each panel aggregates $10{,}000$ Monte Carlo runs. For the empirical distributions, in each run we draw a real-symmetric Wigner matrix ($\beta=1$), append one row and one column to form an $(N{+}1)\times(N{+}1)$ arrowhead matrix, and read the overlaps from its eigenvectors. For the theoretical distributions, we sample the sine--$\beta$ process at $\beta=1$ with Gaussian variables, and evolve the inserted level via the secular equation $S(z)=h$. Evaluating the derivative $S'(z)$ at the corresponding root yields the predicted overlaps according to formula \ref{['eq:bulk-near-anchored']}. Histograms share a common binning and are normalized to unit area.
• Figure 3: Distributions of rescaled overlaps at the right edge for $q=0.9$. Panels show the probability density of $N^{1/3}(\Omega_{i,j}-\delta_{ij})$ for $N=100$ with (left) $(i,j)=(0,1)$ and (right) $(i,j)=(1,2)$. Blue bars show the numerical distributions: in each run we draw a real Wishart matrix with aspect ratio $q=T/N$, augment it by one row and one column, and compute the overlap between the perturbed and unperturbed eigenvectors near the largest eigenvalue. Red bars show the theoretical predictions: we sample the Airy--$\beta$ point process at $\beta=1$ (approximated via GOE), draw independent Gaussian variables $(g_i,g_j)$, and compute the overlaps according to the soft-edge formula in Theorem \ref{['thm:soft-edge-case']} with constant $c_q=(1+\sqrt{q})^{-1/3}$. The agreement validates the soft-edge universality for Wishart matrices. Results are averaged over $10{,}000$ Monte Carlo simulations.
• Figure 4: Distributions of rescaled overlaps at the hard edge for $\alpha=1$. Panels show the probability density of $|\Omega^{(N)}_{i,i-k}|$ for $N=200$ with offset values (top left) $k=-1$, (top right) $k=0$, (bottom left) $k=1$, and (bottom right) $k=2$, corresponding to overlaps between the smallest eigenvectors. Blue bars show the numerical distributions: in each run we draw a real Wishart matrix with rectangularity parameter $\alpha=T-N=1$, augment it by one row and one column, and compute the overlap between the perturbed and unperturbed eigenvectors near the smallest eigenvalue. Red bars show the theoretical predictions: we sample the Bessel point process at $\beta=1$ (approximated via small Wigner eigenvalues with $N_w=1000$, keeping the smallest $80$ points), draw independent Gaussian variables $(g_i)$, and compute the overlaps according to the hard-edge formula in Theorem \ref{['thm:hard-edge-case-wishart']} by solving the secular equation $D_{g^2,\xi,\alpha}(z)=0$ where $D_{g^2,\xi,\alpha}(z)=\sum_i g_i^2/(\xi_i-z)-\chi^2_\alpha/z$. Results are averaged over $15{,}000$ Monte Carlo simulations.

Theorems & Definitions (30)

• Theorem 1: Edge case
• Theorem 2: Bulk case
• Remark 2.1: K-step consequence
• Remark 2.2
• Theorem 3: Left Edge
• Theorem 4: Right edge
• Theorem 5: Hard edge, $q=1$
• Theorem 6: Bulk regime
• Theorem 7: Soft Edge
• Theorem 8: Bulk, real Wishart