Nodal Count for Orthogonally Invariant Ensembles

TL;DR

This work analyzes the nodal count $\phi(A,k)$ of eigenvectors for random real symmetric matrices drawn from orthogonally invariant ensembles, with emphasis on GOE. By expressing $\phi(A,k)$ through sign correlations of quadratic forms on the orthogonal group and reducing to Gaussian space, the authors derive precise asymptotics: $\mathbb{E}_{\Phi,\bm{\Lambda}}[\phi(A,k)] = {n\choose 2}\left( \tfrac{1}{2} + \tfrac{2^{3/2}}{\pi^{3/2}} \mathbb{E}_{\bm{\Lambda}}[\lambda_k] n^{-1/2} + \tilde{O}(n^{-3/2}) \right)$ and $\mathrm{Var}(\phi(A,k)) = \tilde{O}(n^{5/2})$, uniformly in $k$. Under normalization to center the eigenvalues, the normalized nodal count distribution converges to the eigenvalue distribution of the ensemble; in the GOE case this limit is the semicircle law. Consequently, the long-standing Gaussian-nodality conjecture is refuted. The paper also develops a general machinery for transferring nodal statistics to eigenvalue statistics and demonstrates the non-universality of nodal fluctuations, highlighting subtle correlations that defy a universal Gaussian description.

Abstract

We investigate the nodal count of eigenvectors of random matrices interpreted as operators on signed complete graphs. Our focus is on orthogonally invariant ensembles, with particular attention to the Gaussian Orthogonal Ensemble (GOE). We establish that, as the matrix size tends to infinity, the distribution of nodal counts converges to the same limiting law as the eigenvalue distribution. In the GOE case, this limit is the semicircle law. This result refutes a conjecture, motivated by quantum chaos and quantum graphs, which predicted Gaussian behavior of the nodal count.

Figures (3)

• Figure 1: Eigenvalue–nodal count agreement for a Gaussian mixture. Numerical experiment for a single$n\times n$ random matrix $A = O^{\mathsf T}\!\operatorname{diag}(\Lambda)O$ with $n=10^4$, where $O$ is Haar orthogonal and the entries of $\Lambda$ are sampled independently from the Gaussian mixture distribution $f(x) = \tfrac{1}{4\sqrt{2\pi}}\!\left(e^{-\tfrac{(x+5)^2}{2}} + e^{-\tfrac{(x+1)^2}{2}} + 2 e^{-\tfrac{(x-3)^2}{2}}\right)$. We emphasize that this figure shows results for a single sampled matrix, not an average over many realizations.
• Figure 2: Concentration of the nodal count for GOE matrices. In this experiment we sampled $10^6$ independent random GOE matrices of size $n=2^9$. On the left, the thick curve shows the empirical mean $\mathbb{E}[\phi(A,k)/\binom{n}{2}]$, while the thin curves indicate $\pm3$ standard deviations. The mean takes values between $0.48$ and $0.52$, consistent with fluctuations of order $n^{-1/2}$ around $1/2$. On the right, the standard deviation $\mathrm{std}[\phi(A,k)/\binom{n}{2}]$ is plotted as a function of $k$, taking values between $0.925\times10^{-3}$ and $1.075\times10^{-3}$, of order $n^{-1}$, which means $\mathrm{Var}[\phi(A,k)]$ is of order $n^2$.
• Figure 3: Asymptotic statistics of the nodal count for GOE matrices.(A) Log--log boxplot of the 1-Wasserstein distance between $\mathrm{emp}(\phi_{\mathcal{N}}(A))$ and the semicircle distribution, as a function of $\log n$ for $n=2^5,2^6,\dots,2^{13}$. Each box summarizes $100$ independent samples of $\mathrm{GOE}_n$ matrices: the central line indicates the median, the box the interquartile range, whiskers extend to $1.5$ times the interquartile range, and circles mark outliers. (B) Similar log--log boxplot of the Kolmogorov–Smirnov (KS) distance between $\phi(A,k)$ and a Gaussian distribution of the same mean and variance, for $n=2^4,\dots,2^9$. Each box summarizes the distribution of KS distances over all $k\in\{1,\ldots,n\}$, computed from $10^6$ independent GOE matrices. (C) Log--log plot of $\max_k\mathrm{Var}[\phi(A,k)]$ versus $\log n$, estimated from $10^4$ independent samples; the fitted slope $1.998$ and intercept $-3.670$ suggest $\mathrm{Var}[\phi(A,k)]=\tilde{O}(n^2)$.

Theorems & Definitions (20)

• Definition 1.1
• Theorem 1.3
• Corollary 1.4: General convergence
• Corollary 1.5: GOE convergence
• Conjecture 1.6: CLT
• Definition 2.1: Spectral growth bound
• Proposition 2.2: janson1997gaussian
• Lemma 2.3
• proof
• proof : Proof of Corollary \ref{['cor:conver']}