Empirical Eigenvalue Bulk in Random Matrix Theory

• Empirical eigenvalue bulk is the study of non-extremal eigenvalue behavior in large random matrices, characterized by universal laws like the semicircle and Marčenko–Pastur laws.
• It employs diverse ensembles — including Wigner, covariance, and sparse matrices — under various scaling regimes to reveal asymptotic spectral statistics and rigidity.
• The bulk analysis informs practical applications such as signal detection, financial and genomic inference, while ensuring eigenvalue rigidity and delocalization in high dimensions.

The empirical eigenvalue bulk refers to the collective behavior, statistics, and limiting distribution of the non-extremal ("bulk") eigenvalues of large random matrix ensembles or analogous high-dimensional operators. The bulk is distinguished from the spectral edges or outlier eigenvalues and is a principal object of paper in random matrix theory (RMT), high-dimensional statistics, and related fields, revealing universal phenomena such as the semicircle law, Marčenko–Pastur law, and sine-kernel universality. The empirical bulk governs typical eigenvalue statistics in the interior of the spectrum, informing both asymptotic theory and practical inference.

1. Definition, Ensembles, and Scaling Regimes

The empirical eigenvalue bulk describes the limiting behavior of the spectrum’s interior as the matrix size (e.g., $n$, $N$, $p$) tends to infinity with prescribed scaling. Classical examples include:

• Wigner Ensembles: For generalized Hermitian Wigner matrices, the empirical bulk forms under conditions such as matching first four moments, variance scaling $O(1/N)$, and bounded higher moments (Landon et al., 2018). The bulk indices typically exclude $O(N^\varepsilon)$ edge eigenvalues.
• Covariance/Gram Matrices: The bulk is generated from centralized sample covariance matrices $S_{m,n}$, often under aspect ratio limits $m/n \to y \in [A_1, A_2]$, or as $p/n \to \gamma$ for high-dimensional statistics (Dallaporta, 2013).
• Random Regular and Sparse Graphs: In adjacency matrices of $d$-regular graphs ($d\gg (\log N)^C$), the empirical bulk is contained in the set of nontrivial eigenvalues, exhibiting semicircle law statistics (Bauerschmidt et al., 2015). For sparse Erdős–Rényi graphs, centered and rescaled adjacency matrices produce a similar bulk in the regime $pN \gg N^{2/3}$ (Erdos et al., 2011, He, 2019).
• Kernel and Tensor Matrices: Kernel matrices in the polynomial scaling regime, $n \sim p^\ell$, decompose into a bulk part with limiting law given by additive free convolution of semicircle and Marčenko–Pastur laws (Kogan et al., 23 Oct 2024).
• Unitary Truncations and Elliptic Ensembles: Truncations of Haar unitary matrices and elliptic random ensembles yield bulk eigenvalue laws on the unit disc or ellipse, respectively (Meckes et al., 2019, Alt et al., 2021).
• Gaussian β-ensembles at High Temperature: With $\beta_n \sim 2\alpha/n$, the empirical bulk transitions from rigid spectra to Poisson statistics (Duy et al., 2016).

The scaling regime (e.g., polynomial in $p$, sparse degree $d$, fixed $\lim p/n$, high temperature) critically determines the bulk law and variance structure.

2. Limiting Empirical Spectral Laws

The bulk of the empirical spectral distribution (ESD) often converges (a.s. or in probability) to deterministic and universal limiting measures:

• Semicircle Law: For Wigner and regular graph ensembles, the ESD converges to the semicircle law on $[-2, 2]$: $\varrho_{sc}(x) = (1/(2\pi))\sqrt{4-x^2}$ (Bauerschmidt et al., 2015, Erdos et al., 2011).
• Marčenko–Pastur Law: Covariance and Gram matrices yield ESDs converging to $\mu_{MP(y)}$, supported on $[(1-\sqrt{y})^2, (1+\sqrt{y})^2]$ (Dallaporta, 2013, Akama, 2022). In the presence of a one-factor (equi-correlated) model, the scale is modulated by the specific variance, and the support width is $4(1-\rho)\sqrt Q$ for $Q=T/N$ (Akama, 2022).
• Free Convolutions: Kernel matrices in the n $\sim$ p$^\ell$ regime exhibit bulk laws as additive free convolutions: $$\mu_{a,b,\gamma} = (a/\sqrt{\gamma\ell!})(\mu_{mp,\gamma\ell!}-\delta_1) \boxplus \sqrt{b} \mu_{sc}$$, with spectral edges found via cubic equations in the Stieltjes transform (Kogan et al., 23 Oct 2024).
• Deterministic Measures on the Unit Disc/Ellipse: Unitary truncations and elliptic ensembles exhibit bulk laws as explicit radial or area densities ($g_\alpha(z)$ on the disc, $\sigma_\tau$ on the ellipse) (Meckes et al., 2019, Alt et al., 2021).
• Associated Hermite Laws: Gaussian β-ensembles at high temperature ($\beta_n\sim 2\alpha/n$) yield bulk densities $\rho_\alpha(E)$ interpolating between Gaussian and semicircle limits (Duy et al., 2016).

Edges of the bulk are precisely characterized by the limiting law, e.g., $\lambda_\pm = (1\pm\sqrt{\gamma d!})^2$ for tensor kernels (Kogan et al., 23 Oct 2024).

3. Fluctuations and Rigidity in the Bulk

Empirical bulk eigenvalues exhibit strong rigidity and fluctuation bounds:

• Variance Bounds: For covariance matrices, for $\delta n \leq j \leq (1-\delta)n$, $\operatorname{Var}(\lambda_j)\leq C\log n/n^2$ (Dallaporta, 2013). Analogous bounds hold for determinantal point processes arising from truncated unitaries, with $$\operatorname{Var}(\lambda_{(p)})\leq C_\alpha \sqrt{p\log p}/n$$ (Meckes et al., 2019).
• Central Limit Theorems (CLTs): Sparse matrix bulk eigenvalues ($G(N,p)$, $p\in[N^{\varepsilon-1}, N^{-\varepsilon}]$) satisfy CLTs at normalization $N\sqrt{p}$; joint distributions converge to Gaussians with sign-dependent covariance structure (He, 2019).
• Eigenvalue Rigidity: Eigenvalue locations concentrate sharply around their classical locations, with individual deviations decaying subexponentially in $N$ or $m$ (Bauerschmidt et al., 2015, Landon et al., 2018, Meckes et al., 2019).
• Counting Functions and Linear Statistics: Eigenvalue counting functions and linear statistics within the bulk satisfy Gaussian fluctuation bounds at scale $n^{-1/2}$ or $\sqrt{\log n}/n$ (Dallaporta, 2013, He, 2019, Duy et al., 2016).
• Non-Universal Regimes: At high temperature ($\beta_n\to 0$), local bulk eigenvalue statistics transition to Poisson, losing rigidity and level-repulsion (Duy et al., 2016).

4. Bulk Universality and Correlation Functions

Universality is a hallmark of empirical bulk statistics:

• Local Correlation Universality: In random regular graphs and Wigner-type matrices with sufficiently many moments, the $k$-point correlation functions and gap distributions in the bulk converge to those of the GOE/GUE, governed by the sine-kernel determinant (Bauerschmidt et al., 2015, Erdos et al., 2011, Landon et al., 2018).
• Empirical Spacings and Unfolding: After unfolding (i.e., mapping raw eigenvalues via the cumulative equilibrium measure to constant density), the empirical distribution of nearest-neighbor spacings converges in Kolmogorov metric to the Gaudin (sine-kernel) law, at explicit polynomial rates $O(|I_N|^{-1/4+\epsilon})$ for macroscopic intervals (Schubert et al., 2015).
• Dyson Brownian Motion and Moment-Matching: Bulk universality proofs leverage DBM relaxation and four-moment comparison theorems to show local statistics (gaps, correlations) are invariant under broad ensembles (Landon et al., 2018, Erdos et al., 2011).
• Rank-One Perturbations: In factor models (e.g., equi-correlated normal populations), rank-one "spikes" impact only outlier eigenvalues; bulk laws persist under such perturbations due to Bai–Silverstein’s rank-inequality (Akama, 2022).

5. Practical Applications and Inference from the Bulk

Empirical bulk analysis informs a variety of high-dimensional inference and signal detection tasks:

• Spiked Model Factor Estimation: Bulk eigenvalue matching (BEMA) uses bulk quantiles to fit residual variance models, deriving thresholds for detecting the number of spikes/factors in covariance models. BEMA is proven consistent (estimation error $O(n^{-1})$) and provides robust confidence intervals (Ke et al., 2020).
• Financial Data, Genetics, Signal Processing: The Marčenko–Pastur bulk law describes the spectral histogram of sample correlation matrices in finance and genomics (e.g., stock returns, genetic ancestry components), guiding empirical detections (Akama, 2022, Ke et al., 2020).
• Spectral Norms and Extremal Statistics: The empirical bulk determines the convergence of spectral norms to deterministic edges, rigorously bounding operator norms for large kernel and tensor matrices (Kogan et al., 23 Oct 2024).
• Eigenvector Delocalization: Bulk eigenvectors are guaranteed to be fully delocalized (norm $n^{-1/2}$) in random regular graphs, elliptic ensembles, and unitary truncations, precluding concentration in particular directions (Bauerschmidt et al., 2015, Alt et al., 2021).

6. Variations: Low Rank Structure and Perturbations

In extensions beyond classical bulk behavior:

• Kernel/Tensor Decomposition: Random kernel matrices decompose as $K = B + L$, with $B$ carrying the bulk statistics and $L$ low-rank corrections determined by Hermite expansions. The bulk spectrum is characterized by free convolution laws; the low-rank part has negligible impact on bulk statistics for $n \gg p^{\ell-1}$ (Kogan et al., 23 Oct 2024).
• Sparse or Heavy-Tailed Ensembles: Bulk fluctuation regimes may exhibit sign-dependent correlations (sparse matrices), or altered variance scaling, but central limit phenomena remain robust (He, 2019).
• Phase Transitions: In factor models, phase transitions occur as spike strengths vary (Baik–Ben Arous–Péché), controlling whether outlier eigenvalues detach from the bulk or merge into the continuous spectrum (Akama, 2022).

7. Summary Table: Ensemble Classes and Bulk Laws

Ensemble Type	Limiting Bulk Law	Rigidity/Fluctuation Bounds
Wigner, GOE/GUE	Semicircle law + sine-kernel universality	$$\operatorname{Var}(\lambda_j) = O(\frac{\log n}{n^2})$$ (Dallaporta, 2013), CLT (Landon et al., 2018)
Covariance, MP	Marčenko–Pastur ($\gamma=p/n$)	Bulk CLT, Wasserstein $O(\sqrt{\log n}/n)$ (Dallaporta, 2013)
Sparse/Erdős–Rényi	Semicircle via resolvent normalization	Joint CLT, sign-dependent covariance (He, 2019)
Kernel/Tensor, n$\sim$p$^\ell$	Free convolution ($\mu_{a,b,\gamma}$)	Norm converges to deterministic edge (Kogan et al., 23 Oct 2024)
Unitary Truncation	Radial disc law $g_\alpha$	Counting function $\operatorname{Var}\leq C_\alpha i \sqrt{\log i}$ (Meckes et al., 2019)
Elliptic Ensemble	Uniform measure on ellipse	Delocalized eigenvectors (Alt et al., 2021)
High-T β-ensemble	Associated Hermite law, Poisson bulk	CLT for linear statistics, Poisson spacings (Duy et al., 2016)

The empirical eigenvalue bulk thus acts as a universal organizing principle governing both the typical eigenvalue statistics and the structure of fluctuations within large random matrices, with deep connections to universality, rigorous inference, and high-dimensional statistical modeling.