Wilcoxon–Wigner Random Matrices

• Wilcoxon–Wigner random matrices are symmetric matrices constructed from normalized rank statistics of i.i.d. continuous data, offering a fully distribution-free ensemble.
• They display classic spectral properties such as the semicircle law and Gaussian fluctuations for leading eigenvalue and eigenvector due to weak dependence and rank invariance.
• Their parameter-free nature facilitates robust spectral tests for latent community and submatrix detection, outperforming traditional Gaussian-based random matrix models.

Wilcoxon–Wigner random matrices are symmetric random matrices whose entries are structured as normalized rank statistics derived from an independent and identically distributed (i.i.d.) sample of absolutely continuous real variables. This construction yields a matrix ensemble interfacing nonparametric statistics, spectral theory, and hypothesis testing for latent structure in large symmetric data matrices. Their study provides fully parameter-free and distribution-free approaches for spectral detection tasks such as community and principal submatrix detection, overcoming critical limitations of classical Gaussian and moment-based random matrix models, particularly in heavy-tailed or non-Gaussian regimes (Liao et al., 21 Dec 2025).

1. Definition and Ensemble Structure

Given i.i.d. absolutely continuous $(A_{ij}: 1 \le i < j \le n)$, set $N = \tfrac{n(n-1)}{2}$. The normalized rank matrix $\widetilde{R} \in [0,1]^{n\times n}$ is defined entrywise by

$\widetilde{R}_{ij} = \begin{cases} \frac{1}{N + 1} \sum_{1 \le i' < j' \le n} \mathbf{1}\{A_{i'j'} \le A_{ij}\}, & i < j, \[1ex] \widetilde{R}_{ji}, & i > j, \ 0, & i = j. \end{cases}$

For $i<j$, $(N+1)\widetilde{R}_{ij}$ gives the rank of $A_{ij}$ among all off-diagonal entries. The variance of an off-diagonal element is

$$\sigma_n^2 = \mathrm{Var}(\widetilde{R}_{12}) = \frac{1}{12} - \frac{1}{6(N+1)}.$$

The centered, scaled Wilcoxon–Wigner matrix is

$$W_n = \sigma_n^{-1}(\widetilde{R} - \mathbb{E}\widetilde{R}), \qquad \mathbb{E} \widetilde{R}_{ij} = \tfrac{1}{2}, \ i \neq j, \quad W_{ii} = 0.$$

This ensemble differs fundamentally from classical Wigner matrices in that its entries are weakly dependent (via the rank transformation) and exhibit strictly uniform marginal distributions over discrete points, independently of the underlying model beyond continuity (Liao et al., 21 Dec 2025).

2. Main Spectral Properties and Limit Laws

Wilcoxon–Wigner random matrices exhibit global and local spectral universality akin to classical Wigner matrices, including a semicircle law for empirical eigenvalue distributions and operator norm concentration.

Global Semicircle Law

Let $\mu_n$ denote the empirical spectral distribution of $n^{-1/2} W_n$, and let $\mu_{\mathrm{sc}}$ be the semicircle law on $[-2,2]$. For any bounded continuous $f$: $$\int f(x)\, \mu_n(dx) \xrightarrow{a.s.} \int f(x)\, \mu_{\mathrm{sc}}(dx).$$ The operator norm satisfies

$$\frac{\|W_n\|}{\sqrt{n}} \xrightarrow{a.s.} 2, \qquad \Pr\bigl\{\|W_n\| \ge 6\sqrt{n}\bigr\} \leq e^{-c n}$$

for some $c>0$ (Liao et al., 21 Dec 2025).

Fluctuations of Leading Eigenvalue and Eigenvector

Let $\widehat{\lambda}_1(\widetilde{R})$ denote the largest eigenvalue. Define centering and scaling parameters: $$\mu_n = \frac{n-1}{2} + 2\sigma_n^2, \qquad \widetilde{\sigma}_n^2 = \frac{8\sigma_n^4}{n}.$$ The normalized fluctuation of the leading eigenvalue is asymptotically Gaussian: $$\frac{\widehat{\lambda}_1(\widetilde{R}) - \mu_n}{\widetilde{\sigma}_n} \xrightarrow{d} N(0,1).$$ For the leading eigenvector $\widehat{u}_1$ (principal eigenspace), with $u_1 = n^{-1/2} \mathbf{1}_n$, the fluctuation satisfies: $$n\,\widetilde{\sigma}_n^{-1} \Bigl(u_1^\top\widehat{u}_1 - 1 + \frac{1}{6n}\Bigr) \xrightarrow{d} N(0,1).$$ Hence, $u_1^\top\widehat{u}_1\to1$ at rate $n^{-1}$, with $n^{-3/2}$ Gaussian fluctuations (Liao et al., 21 Dec 2025).

3. Proof Sketch and Comparison to Classical Wigner Models

Entrywise, the off-diagonal elements are marginally discrete uniform with order-one variance and diminishing covariance as $n \to \infty$. The proof of the semicircle law exploits that the weak dependence inherent in the rank transformation is negligible: moment methods, Lindeberg replacement, and matrix Bernstein arguments (or symmetrized Hoeffding bounds) establish the convergence.

The Gaussian fluctuation results for the leading eigenvalue and eigenvector rely on perturbative spectral expansions. For the leading eigenvalue, viewing $\widetilde{R} = \mathbb{E}\widetilde{R} + \Gamma$ with rank-one spike structure in $\mathbb{E}\widetilde{R}$, one obtains: $$\widehat{\lambda}_1 - \frac{n-1}{2} = \frac{u_1^{\top} \Gamma^2 u_1}{\lambda_1(\mathbb{E} \widetilde{R})} + \text{(lower order)}$$ with the first-order term vanishing by symmetry; a CLT for the quadratic form yields the stated variance. Analogous expansions drive the eigenvector results (Liao et al., 21 Dec 2025).

Classical Wigner ensembles feature i.i.d. (up to symmetry) entries of mean zero, variance one, and subexponential tails (Erdos, 2010, Tao et al., 2011). Universality, including semicircle law, bulk sine-kernel, and edge Tracy–Widom laws, has been established for such models, often contingent upon four-moment Lindeberg matching (Tao et al., 2011). Wilcoxon–Wigner matrices fall outside this independence paradigm, yet their (weakly dependent) ranks inherit sufficient "asymptotic freeness" for the classical results to transfer under analogous techniques.

4. Parameter-Free, Distribution-Free Spectral Testing

The Wilcoxon–Wigner ensemble yields fully parametrization-free spectral tests for hidden structure based solely on ranks. The test statistic

$$T_n = \frac{\widehat{\lambda}_1(\widetilde{R}) - \frac{n-1}{2} - 2\sigma_n^2}{\widetilde{\sigma}_n}$$

is asymptotically $N(0,1)$ under the null hypothesis (all $A_{ij}$ i.i.d. continuous), and therefore provides a universal basis for latent structure detection.

Community Detection

For a symmetric matrix where nodes belong to two balanced communities, and

$$A_{ij} \sim \begin{cases} F_1, & \theta_i\theta_j = 1, \ F_2, & \theta_i\theta_j = -1, \end{cases}$$

letting $\varepsilon = \mathbb{E}_{X \sim F_1} F_2(X) - 1/2$, the test based on $|T_n|$ is consistent when $|\varepsilon| \gg n^{-1/4}$. This surpasses moment-based methods in robustness, as it incurs no loss under heavy-tailed or unknown $F_1$, $F_2$ (Liao et al., 21 Dec 2025).

Principal Submatrix Detection

For principal submatrix detection (a small $n_1 = o(n)$ subset with different entry distribution), the test is consistent whenever $|\mathbb{E}_{F_1} F_2 - 1/2|\gg n / n_1^{3/2}$. In both settings, no estimation of variances, densities, or moments is required—uniquely positioning this approach among spectral test methodologies (Liao et al., 21 Dec 2025).

5. Comparison with Spiked and Deformed Wigner Ensembles

Classical spiked Wigner models, with entrywise mean $\mu>0$ and variance one, yield a leading eigenvalue centered at $\mu(n-1) + O(1)$ with fixed-order variance—requiring knowledge or estimation of $\mu$ for proper calibration. Furthermore, heavy-tailed entries degrade their statistical performance and render standard universality results inapplicable (Liao et al., 21 Dec 2025, Erdos, 2010).

Wilcoxon–Wigner matrices, leveraging rank invariance, admit universal centering and scaling no matter the underlying distribution, provided only continuity. Thus, the corresponding tests are nonparametric, resilient to outliers or heteroskedasticity, and require no distributional tuning parameters—a stark contrast to spiked or deformed Wigner approaches dependent on explicit moment structure (Liao et al., 21 Dec 2025, Lee et al., 2014).

6. Universality and Relations to Broader Random Matrix Theory

Despite weak dependence among rank-based entries, the spectral behavior of Wilcoxon–Wigner matrices is deeply consonant with the primary universality phenomena of random matrix theory:

• Global and local semicircle law: Empirical spectral distributions converge to Wigner's semicircle law, and local laws hold down to $n^{-1+\varepsilon}$ scales (Liao et al., 21 Dec 2025, Soosten et al., 2019).
• Fine-scale universality: The Four Moment Theorem, local relaxation flow, and Dyson Brownian motion underpinning bulk and edge universality for Wigner matrices continue to apply—implying sine-kernel statistics and Tracy–Widom edge fluctuations under suitable moment conditions (Erdos, 2010, Tao et al., 2011, Lee et al., 2014).
• Nonparametric robustness: The rank-based construction achieves insensitivity to both the entrywise distribution and its moments, provided entries are continuous—aligning the Wilcoxon–Wigner ensemble with the maximal generality of universality results.

7. Applications and Implications

The Wilcoxon–Wigner random matrix ensemble provides a new foundation for high-dimensional, nonparametric, and robust hypothesis testing via spectral methods. Its distribution-free and parameter-free properties offer practical advantages in settings with heavy-tailed, non-Gaussian, or otherwise poorly behaved data, including multivariate analysis, latent community detection, and anomaly or submatrix detection in large graphs. The methodology sidesteps longstanding obstacles in variance estimation and moment dependence, directly leveraging rank invariance in high-dimensional matrix analysis (Liao et al., 21 Dec 2025). In doing so, it establishes a bridge between modern random matrix spectral theory and classical rank-based nonparametrics.